Topic 3 : Linear Equations and Inequalities in One Variable (Straighterline)

The product of complex conjugates

(a +b) (a-bi) = a^2 + b^2

Linear equations have how many answers

1

Quadratic Equations have how many answers

2

Complex Numbers Notes 3

A complex number written in the form a + bi is said to be in standard form. That being said, we sometimes write a − bi in place of a + (−b)i. Furthermore, a number such as 5 + the Square root of 3i is written as 5 + I times the Square root of 3 to emphasize that the factor of i is not under the radical.

Conditional Equation

A conditional equation is true for some values of the variable and false for other values.

Rule of Linear Equation in one variable

A linear equation in one variable is also called a first-degree equation because the degree of the variable term must be exactly one.

Linear Equations in one variable

A linear equation in one variable is an equation that can be written in the form ax + b = 0, where a and b are real numbers, , and x is the variable.

Polynomial equations

A polynomial equation with one side equal to zero and the other side factored as a product of linear or quadratic factors can be solved by applying zero product property.

Quadratic Equation

A quadratic equation can be solved by completing the square and applying the square root property

What would you do to isolate the variable in the equation below, using only one step? x-1/4=7/2

Add 1/4 to both sides of the equation. Adding 1/4 to each side of the equation yields an equivalent equation and isolates the variable: , so x-1/4+ 1/4 =7/2 +1/4

Absolute value equation has no solution

An absolute value equation has no solution if the absolute value expression equals a negative number since an absolute value can never be negative.

|x|=a

An absolute value equation is an equation that contains an absolute value expression. The equation, has two solutions x = a and x = -a because both numbers are at the distance a from 0.

Rational equation

An equation in which each term contains a rational expression. All linear equations are rational equations, but not all rational equations are linear

Contradiction

An equation that is false for all values of the variable is called a contradiction.

Identity

An equation that is true for all values of the variable for which the expressions in an equation are defined is called an identity.

Linear, Compound and Absolute Value Inequalities

An in equality that can be written in one of the following forms is a linear inequality in one variable : ax + b < 0, ax+ b <_ 0, ax + b > 0, or ax + b >- 0

Completing the square

Consider the quadratic equation ax^2 + bx+c = 0 Step 1: Check whether the coefficient of x2, a is 1 or other than 1. Step 2: If , then make it as 1 by dividing each side by a. Step 3: Move the constant term to the right side of the quadratic equation. Step 4: Take one-half of the coefficient of x and square it. Step 5: Add the result obtained in Step 4 to both sides of the equation and complete the square. Step 6: Express the terms in the left side of the equation as a square. Step 7: Simplify the terms in the right side of the equation. Step 8: Equate the result of Step 6 with the result of Step 7. Step 9: Solve the final equation obtained in Step 8 and obtain the required roots.

Divide Complex Numbers

Divide Complex Numbers by multiplying the numerator and denominator by the conjugate of the denominator.

The division property of inequality

Division of both sides of an inequality with a positive number produces an equivalent inequality. If x>y and z>0,then x/z>y/z If x<y and z>0,then x/z<y/z And division on both sides of an inequality with a negative number produces an equivalent inequality if the inequality symbol is reversed. If x>y and z<0,then x/z<y/z If x<y and z<0,then x/z>y/z

Multiplying or Dividing by a Value

Everything is fine if we want to multiply or divide by a positive number: Solve: 3y < 15 If we divide both sides by 3 we get: 3y/3 < 15/3 y < 5 And that is our solution: y < 5 When we multiply or divide by a negative number we must reverse the inequality. Do not try dividing by a variable to solve an inequality (unless you know the variable is always positive, or always negative).

Properties Involving Absolute Value Inequalities

For a real number k > 0, |u| < k is equivalent to −k < u < k. (1) | u | > k is equivalent to u < −k or u > k. (2) Note: The statements also hold true for the inequality symbols ≤ and ≥, respectively.

Addition Property of Equality

For all real numbers a, b, and c: If a = b, then a + c = b + c. If two expressions are equal to each other, and you add the same value to both sides of the equation, the equation will remain equal.

Multiplication Property of Equality

For all real numbers a, b, and c: If a = b, then a • c = b • c (or ab = ac). If two expressions are equal to each other and you multiply both sides by the same number, the resulting expressions will also be equivalent.

Complex Numbers

Given real numbers a and b, a number written in the form a + bi is called a complex number. The value a is called the real part of the complex number and the value b is called the imaginary part.

Complex Numbers Notes 2

If a = 0 and , then a + bi equals bi, which we say is pure imaginary. For Example: The number 0 + 8i is a pure imaginary number and is generally written as simply 8i.

Solving a Linear Equation by Clearing Fractions

If a linear equation contains fractions, it is often helpful to clear the equation of fractions. This is done by multiplying both sides of the equation by the least common denominator (LCD) of all terms in the equation.

Complex Numbers Notes 1

If b = 0, then a + bi equals the real number a. This tells us that all real numbers are complex numbers. Example : 4 + 0i is generally written as the real number 4.

Multiplying ir Dividing by a Negative Number

If both sides of an inequality are multiplied

Zero Product Property

If mn=0, then m=0 or n= 0

Linear Inequalities

If there is a variable in the inequality and its highest power is 1, then then inequality is a linear inequality (unless the variable is in the denominator)

Square Root Property

If x^2 = k, then x= +- the square root of k

Using the Multiplication Property of Equality

Just as you can add or subtract the same exact quantity on both sides of an equation, you can also multiply both sides of an equation by the same quantity to write an equivalent equation. Let's look at a numeric equation, 5 • 3 = 15, to start. If you multiply both sides of this equation by 2, you will still have a true equation. 5 • 3 = 15 5 • 3 • 2 = 15 • 2 30 = 30

Property of Inequality

Let a, b, and c represent real numbers. If x < a, then a > x. If a < b and b < c, then a < c. If a < b and c < d, then a + c < b + d. If a < b, then a + c < b + c and a − c < b − c. a/c < b/c If c is positive and a < b, then ac < bc and a/c > b/c If c is negative and a < b, then ac > bc and . These statements are also true expressed with the symbols >, ≤, and ≥.

Quadratic Equations

Let a, b, c, represent real numbers. A quadratic equation in the varioable x is an equation of the form ax^2 +bx+c =) when a is not equal to zero

Solve Linear Equation with More than one Fraction

Multiply both sides by the LCD of the two denominators

When x is the numerator of a fraction

Multiply both sides by the denominator

When a linear equation contains a variable as denominator

Multiply both sides by the denominator ex. 2= 6/x 2 * x= 6/x * x 2x= 6 2x/2=6/2 x=3

What If I Solve It, But "x" Is On The Right?

No matter, just swap sides, but reverse the sign so it still "points at" the correct value! Example: 12 < x + 5 If we subtract 5 from both sides, we get: 12 − 5 < x + 5 − 5 7 < x That is a solution! But it is normal to put "x" on the left hand side ... ... so let us flip sides (and the inequality sign!): x > 7 Do you see how the inequality sign still "points at" the smaller value (7) ? And that is our solution: x > 7

By definition, i2 = −1, but what about other powers of i?

Notice that even powers of i simplify to 1 or −1. If the exponent is a multiple of 4, then the expression equals 1. If the exponent is even but not a multiple of 4, then the expression equals −1

Solving Inequalities

Our aim is to have x (or whatever the variable is) on its own on the left of the inequality sign: Something like: x < 5 or: y ≥ 11 Example: x + 2 > 12 Subtract 2 from both sides: x + 2 − 2 > 12 − 2 Simplify: x > 10 Solved!

Applications of Quadratic Equations

Quadratic equations are used to model applications with the Pythagorean theorem, volume, area, and objects moving vertically under the influence of gravity

Rational Equations

Rational Equations are solved by multiplying both sides of the equation by the LCD of all fractions in the equations

Problem-Solving Strategy

Read the problem carefully. Determine what the problem is asking for, and assign variables to the unknown quantities. Make an appropriate figure or table if applicable. Label the given information and variables in the figure or table. Write an equation that represents the verbal model. The equation may be a known formula or one that you create that is unique to the problem. Solve the equation from step 3. Interpret the solution to the equation and check that it is reasonable in the context of the problem.

How to Graph a Linear Inequality

Rearrange the equation so "y" is on the left and everything else on the right. Plot the "y=" line (make it a solid line for y≤ or y≥, and a dashed line for y< or y>) Shade above the line for a "greater than" (y> or y≥) or below the line for a "less than" (y< or y≤).

Solving radical equations

Step 1 Isolate the radical. If an equation has more than one radical, choose one of the radicals to isolate. Step 2 Raise each side of the equation to a power equal to the index of the radical. Step 3 Solve the resulting equation. If the equation still has a radical, repeat steps 1 and 2. *Step 4 Check the potential solutions in the original equation and write the solution set. *In solving radical equations, extraneous solutions potentially arise when both sides of the equation are raised to an even power. Therefore, an equation with only odd-indexed roots will not have extraneous solutions. However, it is still recommended that all potential solutions be checked.

Solving a Linear Equation in One Variable Guidelines

Step 1 Simplify both sides of the equation. Use the distributive property to clear parentheses. Combine like terms. Consider clearing fractions or decimals by multiplying both sides of the equation by the least common denominator (LCD) of all terms. Step 2 Use the addition property of equality to collect the variable terms on one side of the equation and the constant terms on the other side. Step 3 Use the multiplication property of equality to make the coefficient of the variable term equal to 1. Step 4 Check the potential solution in the original equation. Step 5 Write the solution set.

Solving a compound inequality

Step 1 To solve a compound inequality, first solve the individual inequalities. Step 2 If two inequalities are joined by the word "and," the solutions are the values of the variable that simultaneously satisfy each inequality. That is, we take the intersection of the individual solution sets. If two inequalities are joined by the word "or," the solutions are the values of the variable that satisfy either inequality. Therefore, we take the union of the individual solution sets.

Substitution

Substitution can be used to solve equations that are in quadratic form

Vertical Position of an object

Suppose that an object has an initial vertical position of s0 and initial velocity v0 straight upward. The vertical position s of the object is given by g -is the acceleration due to gravity. On Earth, g = 32 ft/sec2 or g = 9.8 m/sec2. t-is the time of travel. v0 - is the initial velocity. s0 -is the initial vertical position. s -is the vertical position of the object at time t. Tip:The value of g is chosen to be consistent with the units for position and velocity. In this case, the initial height is given in ft. The initial velocity is given in ft/sec. Therefore, we choose g in ft/sec2 rather than m/sec2.

The addition property of inequality

The addition property of inequality says that adding the same number to each side of the inequality produces an equivalent inequality Ifx>y,thenx+z>y+z Ifx<y,thenx+z<y+z

Solving Linear Inequalities

The aim is to simplify the inequality to get the variable by itself on one side. Whatever is done to one side must be done to both sides

Graphing linear inequalities

The graph of a linear inequality in one variable is a number line. Use an open circle for < and > and a closed circle for ≤ and ≥.

Absolute value equations and inequality

The inequality |x|<2 Represents the distance between x and 0 that is less than 2 or |x|>2 Represents the distance between x and 0 that is greater than 2

The multiplication property of inequality

The multiplication property of inequality tells us that multiplication on both sides of an inequality with a positive number produces an equivalent inequality. Ifx>yandz>0,thenxz>yz Ifx<yandz>0,thenxz<yz

The subtraction property of inequality

The subtraction property of inequality tells us that subtracting the same number from both sides of an inequality gives an equivalent inequality. Ifx>y,thenx−z>y−z Ifx<y,thenx−z<y−z

How to Graph a Linear Inequality

There are three steps: Rearrange the equation so "y" is on the left and everything else on the right. Plot the "y=" line (make it a solid line for y≤ or y≥, and a dashed line for y< or y>) Shade above the line for a "greater than" (y> or y≥) or below the line for a "less than" (y< or y≤).

Solving Inequalities Rules

These things do not affect the direction of the inequality: Add (or subtract) a number from both sides Multiply (or divide) both sides by a positive number Simplify a side But these things do change the direction of the inequality ("<" becomes ">" for example): Multiply (or divide) both sides by a negative number Swapping left and right hand sides

Add or subtract complex numbers

To add or subtract complex numbers, add or subtract their real parts, and add or subtract their imaginary parts. That is,

i^n

To simplify i^n, divide the exponent, n, by 4. The remainder is the exponent of the remaining factor of i once the fourth power of i has been extracted.

To solve a multi-step inequality

To solve a multi-step inequality you do as you did when solving multi-step equations. Take one thing at the time preferably beginning by isolating the variable from the constants. When solving multi-step inequalities it is important to not forget to reverse the inequality sign when multiplying or dividing with negative numbers. Example Solve the inequality −2(x+3)<10 −2x−6<10 −2x−6+6<10+6 −2x<16 −2x−2>16−2 x>−8

Absolute value equations

To solve an absolute value equation isolate the absolute value. Then used one of the following properties for a positive real number k. Solving for u, we have u = k or u = −k. This and three other properties summarized in Let k represent a positive real number. 1. |u| = k is equivalent to u = k or u = −k. 2. |u|= k is equivalent to u = 0. 3.|u| = k has no solution 4. |u| =|w| is equivalent to u = w or u = −w.

Simplifying Expressions in Terms of i

We multiply and divide the square roots of negative real numbers. However, note that the multiplication and division properties of radicals can be used only if the radicals represent real-valued expressions.

How to find the LCD

What is 1/6 + 7/15 The Denominators are 6 and 15: multiples of 6: 6, 12, 18, 24, 30, 36, ... multiples 15: 15, 30, 45, 60, ... So the Least Common Multiple of 6 and 15 is 30. Now let's try to make the denominators the same. Note: what we do to the bottom of the fraction, we must also do to the top For the first fraction we can multiply top and bottom by 5 to get a denominator of 30: × 5 right over arrow 1/6 = 5/30 right under arrow × 5 For the second fraction we can multiply top and bottom by 2 to get a denominator of 30: × 2 right over arrow 7/15 = 14/30 right under arrow × 2 Now we can do the addition by adding the top numbers: 5/30 + 14/30 = 19/30 The fraction is already as simple as it can be, so that is the answer.

Rule for graphing inequalities

When solve standard inequality such as: |6n| < 18 The direction of the inequality does not change. However, when there is a more complex inequality such as : |m-2| < 8 The direction of the inequality changes to the opposite

Absolute value inequality as a compound inequality

When solving an absolute value inequality it's necessary to first isolate the absolute value expression on one side of the inequality before solving the inequality. 2|3x+9|<36 2|3x+9|/2<36/2 |3x+9|<18 −18<3x+9<18 −18−9<3x+9−9<18−9 −27<3x<9 −273<3x3<93 −9<x<3

Using the Addition Property of Equality

When you solve an equation, you find the value of the variable that makes the equation true. In order to solve the equation, you isolate the variable. Isolating the variable means rewriting an equivalent equation in which the variable is on one side of the equation and everything else is on the other side of the equation. When the equation involves addition or subtraction, use the inverse operation to "undo" the operation in order to isolate the variable. For addition and subtraction, your goal is to change any value being added or subtracted to 0, the additive identity. Since subtraction can be written as addition (adding the opposite), the addition property of equality can be used for subtraction as well. So just as you can add the same value to each side of an equation without changing the meaning of the equation, you can subtract the same value from each side of an equation.

Examples of Simplifying Expressions in Terms of i

Write each radical in terms of i first. Terms of i ( If b is a positive real number, then the square root of negative b is equal to the "I" square root B.) Next simplify the radicals or apply the multiplication property of radicals. Then either simplify or write in the form of "i" square root b.

|x+7|=14

You begin by making it into two separate equations and then solving them separately. x+7=14 x+7−7=14−7 x=7 or x+7=−14 x+7−7=−14−7 x=−21

Using the Multiplication Property of Equality with Fractions

You can also multiply the coefficient by the multiplicative inverse (reciprocal) in order to change the coefficient to 1.

Addition property of equality

a = b is equivalent to a + c = b + c. allows one to add the same quantity to both sides of an equation

Multiplication property of equality

a = b is equivalent to ac = bc . allows one to multiply the same quantity by both sides of an equation.

The Discriminant to the equation

ax^2 + bx + c = 0 when a is not equal to zero is given by b^2 -4ac. The discriminant indicates the number of and type of solutions to the equation. if b^2 -4ac < 0, the equation has 2 nonreal solutions if b^2 -4ac = 0, the equation has 1 real solution if b^2 - 4ac > 0, the equation has 2 real solutions.

5-8|-2n|=-75

get the absolute value by itself. 5-8|-2n|=-75 5-5-8|-2n|=-75-5 8|-2n|/ 8=-80/-8 |-2n|=10 -2n= 10 or -2n=-10 -2n/-2 = 10/-2 or -2n=-10/-2 n=-5 or n=5 {-5,5}

|v+8|-5=2

get the absolute value by itself. |x+8|-5+5=2+5 |x+8| = 7 v+8= 7 or v+8=-7 v+8=7-8 or v+8-8= -7-8 v=-1 or v=-15

>

greater than

≥

greater than or equal to

The Imaginary Number i

i=the square root of and i2 = −1 If b is a positive real number, then .

<

less than

≤

less than or equal to

The absolute number of a number a is written as

|a| And represents the distance between a and 0 on a number line.

Solving Two Inequalities

−2 < 6−2x/ 3 < 4 First, let us clear out the "/3" by multiplying each part by 3. Because we are multiplying by a positive number, the inequalities will not change: −6 < 6−2x < 12 Now subtract 6 from each part: −12 < −2x < 6 Now multiply each part by −(1/2). Because we are multiplying by a negative number, the inequalities change direction. 6 > x >−3 And that is the solution! But to be neat it is better to have the smaller number on the left, larger on the right. So let us swap them over (and make sure the inequalities point correctly): −3 < x < 6