College Algebra WGU CIC1

Slope of a Line

- Measures the steepness every nonvertical line - Positive slope: Line goes up from left to right - Negative slope: Line goes down from left to right - Zero slope: Horizontal Line - Undefined slope: Vertical Line

whole number

0,1,2,3,4...

Subtracting Polynomials

1. In the set that is being subtracted: Reverse the minus and plus signs within the parenthesis. 2. In column format perform the calculations. 3. Remove any solutions with "0"

Solving Linear Equations with Fractions

1. Multiply each side by the reciprocal of the fraction. 4a/5 = 12 The reciprocal of the fraction 4/5 = 5/4 5/4 numerator × 4a/5 = 5/4×12 2. Reduce and simplify. a. 5/4 times 4/5 = 0 so you are left with "a" for the left side. b. 5/4 times 12 = cross reduce: the denominator of 4 goes into 12 three times and the 4 is cancelled out. Reduces go 5 times 3. Solution: a=15

Inverse Functions

1. Only one to one functions have inverses If g is the inverse of f then f is the inverse of g. We say f and g are inverses of each other. 2. If f and g are inverses of each other then both are one to one functions. 3. f and g are inverses of each other if and only if: (f₀g)(x) = x , x in the domain of g and (g₀ f)(x) = x , x in the domain of f

Adding Polynomials

1. Place all like terms together. (all x's; all yx's) 2. Add all like terms.

Word problem tips

1. Read the problem entirely before trying to solve anything. Get a good feeling of the problem and try to see what information is available and what is missing. 2. Organize the information by naming the problem. Use variables for the unknown. 3. Look for the key words to determine what mathematical action to use. 4. BE CAREFUL WITH "LESS THAN": Be sure to use real world translations of the word problem. Example: 2.5 less than x does not mean: 2.5 - x.... If you are asked: Tony makes $2.5 per hour less than Tina. Do not subtract Tina's wage from Tony's; instead subtract $2.50 from Tina's wage. "less than" construction is backwards" 5. Order is important in the "quotient/ratio of" and "difference between/of" constructions. If a problems says "the ratio of x and y", it means "x divided by y", not "y divided by x". If the problem says "the difference of x and y", it means "x - y", not "y - x". 6. "How much is left" word problems: You will be given some total amount. Smaller amounts, of unspecified sizes, are added (combined, mixed, etc) to create this total amount. You will pick a variable to stand for one of these unknown amounts. After having accounted for one of the amounts, the remaining amount is whatever is left after deducting this named amount from the total.

Solving Linear Equations with Integer Coefficients

1. Simplify the equation by adding together all of the "x" values. For example, the equation x + 3x + 8 = 5² + 3 can be simplified by adding the x values on the left side of the equation: x + 3x = 4x. The equation becomes 4x + 8 = 5² +3. 2. Calculate all the numbers on both sides of the equation. For example, the 5² + 3 value in the equation 4x + 8 = 5² +3 can be simplified by finding the square of 5, then adding 3 to that: 5² + 3 = 25 + 3 = 28. The equation becomes 4x + 8 = 28.

Geometry Word Problems

1. You need to know the basics of geometric formulas: Rectangles: a. Area: length times the width b. Perimeter means "length around the outside". 2×l + 2×w Triangles: a. Area: (1/2)bh b. Perimeter: The sum of all three sides. Circles: a. Area: (π)p² b. Circumference: 2(π)p c. Radius: Distance from the center to the outside of the circle. d. Diameter: The length all the way across the circle

natural number

123456789

Complex Numbers

A combination of a "real" number and an "imaginary" number. Imaginary Numbers: When squared, they give a negative result. We have to imagine that there is a number because we need it. i=√⁻1 A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, which satisfies the equation i2 = −1. In this expression, a is the real part and b is the imaginary part of the complex number.

Definition of a Function

A function is a relation such that for each element in the domain, there is exactly one corresponding element in the range. There are no duplication of x

Vertical Line Test

A graph in the Cartesian plane is the graph of a function if and only if no vertical line intersects the graph more than once.

Linear Expressions

A linear equation in one variable is an equation that can be written in the form ax + b = c, where a, b, and c are real numbers and a ≠ 0. A linear expression is an expression with a variable in it; however, the variable is only raised to the first power. For example, 5a is a linear expression, because it is understood that a is raised to the first power. Any pattern of numbers that is increasing or decreasing by the same amount every step of the way. This means that the only two things that we need to define a linear equation are where the pattern begins and what that pattern moves by. What that leaves us with is the slope-intercept form of the linear equation, y = mx + b, where the m value is the slope, and the b value is the y-intercept.

Exponential Notation

A method in mathematics that allows the representation of numbers in shorter form; how many times to use the number in a multiplication.

Simplify any radical expressions that are perfect cubes

A perfect cube is the product of any number that is multiplied by itself twice, such as 27, which is the product of 3× 3 × 3. To simplify a radical expression when a perfect cube is under the cube root sign, simply remove the radical sign and write the number that is the cube root of the perfect cube. For example, 512 is a perfect cube because it is the product of 8 × 8 × 8. Therefore, the cube root of the perfect cube 512 is simply 8.

Simplify any radical expressions that are perfect squares

A perfect square is the product of any number that is multiplied by itself, such as 81, which is the product of 9 x 9. To simplify a radical expression that is a perfect square, simply remove the radical sign and write the number that is the square root of the perfect square. For example, 121 is a perfect square because 11× 11 is 121. You can simply remove the radical sign and write 11 as your answer.

Solving Linear Equations with Radicals

A radical expression is an equation in which at least one variable expression is stuck inside a radical, usually a square root. √x When you have a variable inside a square root, you undo the root by doing the opposite: squaring. For instance, given sqrt(x) = 4, you would square both sides: 4 sqrt = 16 Solution: x = 16

Definition of a Relation

A relation is a correspondence between two sets A and B such that each element of set a corresponds to one or more elements in set B Domain: Elements in Set A Range: Elements in Set B

Interval notation

A shorthand way of writing intervals using parentheses and brackets. x+4<4x-5 . Subtract 4 from both sides: x<4x-9 . Subtract 4x from both sides: -3x<-9 .Multiply both sides by -1 (reverse the inequality): (-1)(-3x)>(-2)(-9) 3x>9 Solution: x>3 Interval Notation: (3,∞)

integer

A whole number (not a fraction) that can be positive, negative, or zero

Zero Exponent Rule

ANY BASE RAISED TO THE ZERO POWER IS EQUAL TO 1

Multiplying and Dividing Exponential Terms

Add exponents when multiplying two exponential terms; subtract when dividing. This concept can also be used to simplify variable expressions. Example: 6x³ × 8x⁴ + (x¹⁷ - 15) (6 × 8)³ + 4 + (x ¹⁷ - 15) 48x7 + x2

Rational Expressions

An algebraic expression that can be written as a fraction whose numerator and denominator are polynomials and the denominator does not equal "0"

Horizontal and Vertical Lines

Any horizontal line that contains the point (a,b) the equation of that line is y = b and the slope is m = 0. Any vertical line that contains the point (a,b) the equation of that line is x = a and the slope is undefined. Given the point (-5,12) The equation of the horizontal line containing that point is y = 12. The equation of the vertical line containing that point is x = -5.

rational numbers

Any number that can be written as a fraction using only integers.

Exponents: Zero Power

Anything raised to the zero power is one.

Step #2 Simplifying Expressions: Simplify the Expression

Construct an expression from your new, smaller set of terms. Get a simpler expression that has one term for each different set of variables and exponents in the original expression. This new expression is equal to the first. New expression is 6x - 2. This simplified expression is equal to the original (1 + 2x - 3 + 4x), but is shorter and easier to manage. It's also easier to factor.

Solving Linear Equations with Decimals

Decimal numbers are fractions in disguise, "clear the decimal" in equations with decimal numbers. Count the largest number of digits behind each decimal point and multiply both sides of the equation by 10 raised to the power of that number. Example: 0.25 x + 0.6 = 0.1 Because there are two digits behind the decimal in 0.25, we need to multiply both sides of the equation by 10² = 100. (distribute the 100 inside the parentheses). 1. 25 x + 60 = 10 2. Isolate the x by subtracting 60 from both sides: 25x = -50 3. Divide the right side by the left side -50/25x Solution: x = -2

Factoring #2: Dividing the terms by the GCF

Divide every term in your equation by the greatest common factor. The resulting terms will all have smaller coefficients than in the original expression. Factor out 9x ² + 27x -3 by its greatest common factor of 3. To do so divide each term by 3. 1. 9x² divided by 3 = 3x² Equals: 27x divided by 3 = 9x Equals: -3 divided by 3 = -1 Thus, our new expression is 3x2 + 9x - 1.

Dividing Polynomials

Dividing polynomials using long division: The numerator (top number) is divided by the denominator (bottom number. -Divide the first term of the numerator by the first term of the denominator, and put that in the answer. -Multiply the denominator by that answer, put that below the numerator -Subtract to create a new polynomial -Repeat, using the new polynomial

Determining the Domain of a Function Given the Equation

Domain is the set of all values of x for which the function is defined. A number x = a is in the domain of a function ƒ if ƒ(a) is a real number. The domain of every polynomial function is all real numbers

Determine the domain and range of a function from its graph

Domain: All the x-values Range: All the y-values List all values without duplication y= - 4⁴ + 4

Simplify a radical expression with variables and numerals that is a perfect square

EXAMPLE: √36a² take apart the expression by first looking for perfect squares in the numbers, and then looking for perfect squares in the variables. Then, remove the radical sign and let their square roots remain. the square root of 36 x a squared. 1. 36 is a perfect square because 6 × 6 = 36. 2. a squared is a perfect square because a times a is a squared. Now that you've broken down these numbers and variables into their square roots, simply remove the radical sign and leave the square roots. The square root of 36 ×∧ a 6a.

factoring trinomials into two binominals without a leading co-efficient of one

Example: 2x² - 18x + 40 is there a greatest common factor among all of our terms, and we see that 2 is a greatest common factor. Factoring out the 2 leaves you with: 2(x²-9x+20) Now apply the two step process to factor what is inside the parenthesis. - Now look for two numbers that when multiplied together equal 20. Also those two numbers when added together must equal -9 (the middle number). Solution: -4 and -5 multiplied together = 20 AND -4+ -5 = =9 Trinomial factors into: 2(x-9)(x-4)

Use square factors to simplify radicals

Expressions under a square root sign are called radical expressions. Simplify by identifying square factors and perform the square root operation on these separately to remove them from under the square root sign. Example: √(90) If we think of the number 90 as the product of two of its factors, 9 and 10, we can take the square root of 9 to give the whole number 3 and remove this from the radical. In other words: √(90) √(9 × 10) (√(9) × √(10)) 3 × √(10) 3√(10)

Factoring Polynomials Example

Factor 3x - 12: The only thing common between the two terms (that is, the only thing that can be divided out of each term and then moved up front) is a "3". 1. So factor this number out to the front: 3x - 12 = 3( ) 2. When the "3" is divided out of the "3x", you are left with only the "x" remaining. Put that "x" as the first term inside the parentheses: 3x - 12 = 3(x ) 3. When "3" is divided out of the "-12", a "-4" was left behind, so put that in the parentheses, too: 3x - 12 = 3(x - 4) 4. Final answer: 3(x - 4) Do not to drop "minus" signs when you factor.

Factoring Polynomials

Factoring polynomial expressions is not the same but similar to factoring numbers. When factoring numbers or factoring polynomials, you are finding numbers or polynomials that divide out evenly from the original numbers or polynomials. But in the case of polynomials, you are dividing numbers and variables out of expressions, not just dividing numbers out of numbers. Simple factoring in the context of polynomial expressions is backwards from distributing. That is, instead of multiplying something through a parentheses, you will be seeing what you can take back out and put in front of a parentheses, such as: You need to see what can be factored out of every term in the expression. Don't make the mistake of thinking that "factoring" means "dividing something off and making it magically disappear". Factoring means "dividing out and putting in front of the parentheses". Nothing "disappears" when you factor; things merely get rearranged.

Finding x intercepts

Find the x- and y-intercepts of 25x² + 4y² = 9 "x-intercept(s):" y = 0 for the x-intercept(s), so: 25x² + 4y² = 9 25x² + 4(0)² = 9 25x² + 0 = 9 x² = 9/25 x = ± ( 3/5 ) Then the x-intercepts are the points (⁻3/5, 0) and (⁻³/₈, 0)

Finding Y intercepts

Find the y-intercepts of 25x² + 4y² = 9 x = 0 for the y-intercept(s), so: 25x² + 4y² = 9 25(0)² + 4y² = 9 0 + 4y² = 9 y² = 9/4 =y ± ( 3/2 ) Then the y-intercepts are the points (0, 3/2) and (0, ⁻3/2)

Multiplying Polynomials

First multiply the constants, then multiply each variable together and combine the result Example: (2xy)(4y) Equals: 2×4×xy×y Equals: 8xy (and y is to the 2nd power)

Point-slope Form of the Equation of a Line

Given the slope of a line M and a point of the line (x₁, y1), the point-slope form of the equation of a line is given by: y - y¹ = m(x -x₁) You must know the slope of the line a and a point on the line.

Slope-Intercept Form of the Equation of a Line

Given the slope of a line m and the y-intercept, b, the slope-intercept form of the equation of a line is given by: y=mx+b

Factoring: #1 Identify the greatest common factor

Identify the greatest common factor in the expression. Factoring is a way to simplify expressions by removing factors that are common across all the terms in the expression. To start, find the greatest common factor that all of the terms in the expression share - in other words, the largest number by which all the terms in the expression are evenly divisible. Example: 9x to the power of 2 + 27x - 3. Every term in this equation is divisible by 3. Since the terms aren't all evenly divisible by any larger number, we can say that 3 is our expression's greatest common factor.

Step #1 Simplifying Expressions: Identify and Combine Like terms

Identify: Like terms have the same configuration of variables, raised to the same powers. They must have the same variable or variables, or none at all, and each variable must be raised to the same power, or no power at all. Example: 1 + 2x - 3 + 4x. Combine: Add terms together (or subtract in the case of negative terms) to reduce each set of terms with the same variables and exponents to one singular term. Example: 2x and 4x and 1 and -3 are like terms 1 + -3 = -2 2x + 4x = 6x 1 + -3 = -2

Commutative Property of Addition

If a and b are real numbers, then a + b= b+a Example: 3+5=8 and 5+3=8

Commutative Property of Multiplication

If a and b are real numbers, then ab = ba Example: (-4)(6)=-24 and (6)(-4)=-24

Distributive Property

If a,b, and c are real numbers, then a(b+c) = ab + ac Example: 3(x-4) = 3× x + 3 × (-4) = 3x - 12

Associative Property of Multiplication

If a,b,and c are real numbers, then (a+b)+c=a+(b+c) Example: (-8×-2)×3 = 16×3=48 and -8×(-2×3)=-8×(-6)=48

Definition of Slope

If x₁ ≠ x₂ the slope of a nonvertical line passing through distinct points(x₁y₁) and (x₂ x₂) is given by: m= rise/run = change in y/change in x = y₂-y₁/x₂-x₁

Word Problem: Age

In February of the year 2000, Tina was one more than eleven times as old as my son Raymond. In January of 2009, I was seven more than three times as old as him. How old was my son in January of 2000? Here's how you'd figure out his age: 1. Name the items and translate into math: Let "E " stand for Tina's in 2000, and let "R " stand for Raymond's age. Then E = 11R + 1 in the year 2000 (from "eleven times as much, plus another one"). In the year 2009 (nine years after the year 2000), Raymond and Tina will each be nine years older, so our ages will be E + 9 and R + 9. Also, Tina was seven more than three times as old as Raymond was. So: E + 9 = 3(R + 9) + 7 = 3R + 27 + 7 = 3R + 34. This gives you two equations, each having two variables: ADVERTISEMENT E = 11R + 1 E + 9 = 3R + 34 If you know how to solve systems of equations, you can proceed with those techniques. Otherwise, you can use the first equation to simplify the second: since E = 11R + 1, plug "11R + 1 " in for "E " in the second equation: E + 9 = 3R + 34 (11R + 1) + 9 = 3R + 34 11R - 3R = 34 - 9 - 1 8R = 24 R = 3 Remember that the problem did not ask for the value of the variable R; it asked for the age of a person. So the answer is: Richard was three years old in January of 2000.

Function Notation

Instead of using the variable y, letters such as f, g, or h (and others) are commonly used for functions. If we want to name a function f, then for any x-value in the domain, we call the y-value (or function value) f(x). f(x) is read "f of x" or "the value of the function f at x"

Distance Word Problems

Involve something travelling at some fixed and steady ("uniform") pace ("rate" or "speed"), or else moving at some average speed. Whenever you read a problem that involves "how fast", "how far", or "for how long", you should think of the distance equation, d = rt, where d stands for distance, r stands for the (constant or average) rate of speed, and t stands for time. Be sure that the units for time and distance agree with the units for the rate. For instance, if they give you a rate of feet per second, then your time must be in seconds and your distance must be in feet. Sometimes you may be tricked by using the wrong units, and you have to catch this and convert to the correct units.

Factoring to Simplify Fractions

Let's say our original example expression, 9x2 + 27x - 3, is the numerator of a larger fraction with 3 in the denominator. This fraction would look like this: (9x2 + 27x - 3) ÷ 3. We can use factoring to simplify this fraction. Let's substitute the factored form of our original expression for the expression in the numerator: (3(3x2 + 9x - 1)) ÷ 3 Notice that now, both the numerator and the denominator share the coefficient 3. Dividing the numerator and denominator by 3, we get: (3x2 + 9x - 1) ÷ 1. Since any fraction with "1" in the denominator is equal to the terms in the numerator, we can say that our original fraction can be simplified to 3x2 + 9x - 1.

Exponents: Quotient with a Negative Power

Negative exponents signify division - find the reciprocal of the base. When a denominator is raised to a negative power, move the factor to the numerator, keep the exponent but drop the negative. 1/5⁻³ (with five to the -3rd power) = 5³

Exponents: Negative power

Negative exponents signify division. In particular, find the reciprocal of the base. When a base is raised to a negative power, reciprocate (find the reciprocal of) the base, keep the exponent with the original base, and drop the negative.

Linear Expressions: Converting verbal statements into mathematical statements

Read the question carefully and recognize the key words as: 1. Add: -Sum; More than; increased by; combined, together; total of; added to 2. Subtract: Decreased by; minus, less; difference between/of; less than, fewer than 3. Multiply: Of; times, multiplied by; product of; increased/decreased by a factor 4. Divide: per,a; out of; ratio of, quotient of; percent (divide by 100) 5. Equals: is, are, was, were, will be; gives; yields; solve for

Simplifying Expressions

Reducing an expression by combining like terms and using order of operations.

Exponents

Shorthand for repeated multiplication. The rules for performing operations involving exponents allow you to change multiplication and division expressions with the same base to something simpler.

Rational Expressions Example

Simplify the following expression: (2x) ÷ (x²) 1. Cancel off any common numerical or variable factors. The numerator factors as (2)(x); the denominator factors as (x)(x). Anything divided by itself is just "1", so cross out any factors common to both the numerator and the denominator. Considering the factors in this particular fraction, you get: 2x÷ x² = (2x) ÷ (xx) = 2 / x You can only cancel factors, not terms!

Solving Linear Equations

Solve x + 6 = -3 Get the x by itself; that is, I want to get "x" on one side of the "equals" sign, and some number on the other side. I want just x on the one side, since the 6 is added to the x, you need to subtract ab 6 from the x in order to "undo" having added a 6 to it. No matter what kind of equation you're dealing with -- linear or otherwise -- whatever you do to the one side, you must do the exact same thing to the other side! Example: x + 6 = -3 -6 -6 ___________ x = -9 The same "undo" procedure works for subtraction:

Coin word problems

TIPS: 1. Convert the relationships between the numbers of coins (if given) into equations 2. Convert the statements about the values of the coins (if given) into equations that state the values all in the same unit (for instance, in cents). 3. Label everything! EXAMPLE: Sue has $1.15 in nickels and dimes, totally 16 coins. How many nickels and how many dimes does Sue have? Make "n" equal the number of nickels. Then the number of dimes is equal to 16-n. So, Sue has 5 ×2n cents in nickels and 10×(16-n) cents in dimes. Since the total amount Sue has is equal to $1.15, or 115 cents, this leads to the equation 5n+10×(16-n)=115 Simplify this equation: 5n+160-10n=115 (after brackets opening) -5n=115-160 (after collecting like terms at the left side and moving 160 to the right side with the opposite sign) -5n+=-45 (after collecting like terms at the right side) n=9 (after dividing both sides by -5). The number of nickels is equal to 9. The number of dimes is equal to 16 minus the number of nickels, that is 16-n = 16-9 = 7 ANSWER: 9 Nickles and 7 dimes

absolute value

The distance of a number from zero on a number line

Find a perfect square in the variable

The square root of a to the second power would be a. The square root of a to the third power is broken down into the square root of a squared times a -- this is because you add exponents when you multiply variables, so that a squared times a reverts back to a cubed. Therefore, the perfect square in the expression a cubed is a squared.

Standard Form Equation of a Line

The standard form of an equation of a line is given by Ax + By = C, where A, B, and C are real numbers such that A and B are both not zero. This text always includes nonfractional coefficients and A is always greater than or equal to zero. Eliminate fractions by multiplying the equation by the LCD.

Number Word Problems

The sum of two consecutive integers is 15. Find the numbers. What you know: 1. You are adding two numbers 2. The sum is 15 3. The second number is one more than the first (consecutive integers) Represent the first number by "n" and the second number will be "n = 1". Their sum is: n + (n + 1) = 15 2n + 1 = 15 2n = 14 n = 7 To answer the question "what are the two numbers that equal 15" : The numbers are 7 and 8

Polynomial

The sum or difference of one or more monomials. A polynomial with two terms is a binomial. With three terms is a trinomial The degree of a polynomial is the degree of the highest monomial term. A polynomial can have constants, variables and exponents, but never division by a variable. Can have just one constant. In graphing the lines are smooth and continuous. The standard form for writing a polynomial is to put the terms with the highest degree first

Word Problem Example - Painting a Fence

Tina can paint a fence in 12 hours. Tony helps and they paint the fence in one hour. How fast can Tony paint the fence by himself? X represents how fast Tony can paint the fence by himself: 1/X Tina can paint the fence by herself in 12 hours. So she can complete 1/12 of the painting in one hour. Working together it only took one hour. Time needed Done in 1 hour to complete job Tina 12 1/12 Tony x 1/x Together 1 1 Create an expression adding Tina and Tony's time together: 1/12 + 1/x =1 Solve for X: Determine the LCD (which is 12x). Multiply by 12X to clear the fractions. 12x(1/12 = 1/x) = 12x(1) Simplify: x+12 = 12x Subtract x from both sides: x + 12 -x = 12x - x = x = 11 12/11 : 11x/11 Divide both sides by 11: 12/11 = x FINALLY: 12/11 = 1 1/11 hours for Tony to paint by himself

Determining Whether Equations Represent Functions

To determine whether an equation represents a function, we must show that for any value in the domain, there is exactly one corresponding value in the range. To make this determination, solve for y. If the solution indicates there are two possible y-values for a given x-value, it is not a function.

Factoring #3: Make new expression equal to the old one

To make our new expression equal to the old one, we'll need to account for the fact that it has been divided by the greatest common factor. Enclose your new expression in parentheses and set the greatest common factor of the original equation as a coefficient for the expression in parentheses. For the example expression, 3x² + 9x - 1: Enclose the expression in parentheses and multiply by the greatest common factor of the original equation to get 3(3x (² + 9x - 1). This equation is equal to the original, 9x² + 27x - 3.

Twice the larger of two numbers

Twice the larger of two numbers is three more than five times the smaller, and the sum of four times the larger and three times the smaller is 71. What are the numbers? The point of exercises like this is to give you practice in unwrapping and unwinding these words, and turning the words into algebraic equations. The point is in the solving, not in the relative "reality" of the problem. That said, how do you solve this? The best first step is to start labelling: the larger number: x the smaller number: y twice the larger: 2x three more than five times the smaller: 5y + 3 relationship between ("is"): 2x = 5y + 3 four times the larger: 4x three times the smaller: 3y relationship between ("sum of"): 4x + 3y = 71 Now I have two equations in two variables: 2x = 5y + 3 4x + 3y = 71 I will solve, say, the first equation for x: x = (5/2)y + (3/2) Then I'll plug the right-hand side of this into the second equation in place of the "x": 4[ (5/2)y + (3/2) ] + 3y = 71 10y + 6 + 3y = 71 13y + 6 = 71 13y = 65 y = 65/13 = 5 Now that I have the value for y, I can solve for x: x = (5/2)y + (3/2) x = (5/2)(5) + (3/2) x = (25/2) + (3/2) x = 28/2 = 14 As always, I need to remember to answer the question that was actually asked. The solution here is not "x = 14", but is the following sentence: The larger number is 14, and the smaller number is 5.

Step #3 Simplifying Expressions: Order of operations

Use the acronym PEMDAS to remember the order of operations. -Parentheses -Exponents -Multiplication -Division -Addition -Subtraction

Complex Numbers in Standard Form example

What is the standard form of (3 - √ -5)(2 + √-10) ? (3 - i√5)(2 + i√10)=(6 + √50) + i(3√10 - 2√5)

Exponents: Quotient with the same base

When dividing like bases, keep the base the same and subtract the denominator exponent from the numerator exponent.

Exponents: Product with the same base

When multiplying like bases, keep the base the same and add the exponents.

Exponents: Power to a power

When raising a base with a power to another power, keep the base the same and multiply the exponents.

Exponents: Quotients to a power

When raising a fraction to a power, distribute the power to each factor in the numerator and denominator of the fraction.

Exponents: Product to a power

When raising a product to a power, multiply the power to each factor.

Percent of Word Problems

When you learned how to translate simple English statements into mathematical expressions, you learned that "of" can indicate "times". This frequently comes up when using percentages. If you need to find 16% of 1400, you first convert the percentage "16%" to its decimal form; namely, the number "0.16". (When you are doing actual math, you need to use actual numbers. Always convert the percentages to decimals!) Then, since "sixteen percent OF fourteen hundred" tells you to multiply the 0.16 and the 1400, you get: (0.16)(1400) = 224. This says that 224 is sixteen percent of 1400. Percentage problems usually work off of some version of the sentence "(this) is (some percentage) of (that)", which translates to "(this) = (some decimal) × (that)". You will be given two of the values, or at least enough information that you can figure two of them out. Then you'll need to pick a variable for the value you don't have, write an equation, and solve for that variable.

Interval

a set containing all real numbers or points between two given real numbers or points

radical expression

an algebraic expression that includes a root. The root may be a square root, a cube root, or any other power. Simplifying a radical expression can help you solve an equation. Simplifying radical expressions involves removing the root when possible, or reducing the radicands, the numbers inside the radical symbol, as much as you can.

factoring trinomials into two binominals with a leading co-efficient of one

first look at the constant on the end, and you want to find two numbers that multiply to equal that constant but that also add up to be the middle co-efficient. These are going to be the numbers that go into your parentheses. Example: x²+×−12 . The constant on the end is -12 . Pick any two numbers that when multiplied together equal 4 x -3=-12 (check to see if the two numbers picked when adding together equal the middle number of the equation (which is 1). . 4 + -3 = 1 so the numbers chosen are correct. . Solution: (×+4)(×-3)

irrational numbers

numbers that cannot be expressed in the form a/b, where a and b are integers and b =0.

Break down an imperfect radical expression into its multiples

step 1: The multiples are the numbers that multiply to create a number -- for example, 5 and 4 are two multiples of the number 20. To break down an imperfect radical expression by its multiples, write down all of the multiples of that number (or as many as you can think of, if it's a large number) until you find one that is a perfect square. For example, all the multiples of the number 45: 1, 3, 5, 9, 15, and 45. 9 is a multiple of 45 that is also a perfect square. 9 × 5 = 45. STEP 2: Remove any multiples that are a perfect square out of the radical sign: 9 is a perfect square because it is the product of 3 × 3. Take the 9 out of the radical sign and place a 3 in front of it, leaving 5 under the radical sign.

real numbers

the set of rational numbers and irrational numbers

x and y intercepts

x-intercept is a point in the equation where the y-value is zero y-intercept is a point in the equation where the x-value is zero Whichever intercept you're looking for, the other variable gets set to zero. Also: think of the following terms interchangeably: "x-intercepts" = "roots" = "solutions" = "zeroes"

Simplify a radical expression with variables and numerals that is not a perfect square

√ 50a³ break down the expression into numerals and variables, and search for perfect squares within the multiples of both. Then, pull any perfect squares out of the radical expression. -Break down 50 to find any multiples that are a perfect square. 25 × 2 = 50 and 25 is a perfect square because 5 × 5 = 25. To simplify root 50, you can pull a 5 out of the radical sign and leave 2 within it. -Break down "a" to the third power to find any multiples that are a perfect square. a to the third power is really a squared times a, and a squared is a perfect square. You can pull one a out of the radical sign and leave one a inside the sign. Therefore, root a cubed is really a root a. -Put it all together. Just place everything you took out of the radical sign out of the radical sign and keep everything that you kept in the sign in the sign. Combine 5 root 2 and a root a to create 5 x a root 2 x a.