Praxis: Math (Algebra thinking)

Keisha had $45. She went to work, where she earned an hourly wage of $13.50 per hour. Write an equation representing Keisha's total amount of money after working a certain number of hours.

$ before work + $ earned at work = total money (m). 45 + 13.50h = M

For example, "twice the sum of a number x and 3" translates to

2(x+3). The parentheses indicate that the entire sum is doubled. Without the parentheses only the x gets doubled: 2x+3

Inequalities: Is Less Than

<, all the numbers to the left of that point on the number line x<-1

It is often easiest to write a linear equation or inequality in slope-intercept form

(y = mx + b) where m is the slope and b is the y-intercept. Consequently, one way to write an equation that represents a linear relationship is to identify which number represents the slope and which number represents the y-intercept, and then substitute the values into the slope-intercept form equation.

For example, "the product of 2 and the quantity 5x minus 10" translates to

2(5x−10). The use of the word quantity indicates that the entire expression of 5x-10 should be multiplied by 2. Without the parentheses, 2⋅5x-10, only the 5x term gets multiplied.

Simplify the expression: 4(7x-3)

28x−12

In the expression 2-( 3x+4), the subtraction sign in front of the parentheses still means that every term inside the parentheses must get subtracted from 2. It is simplified as follows:

2−(3x+4) 2 - 3x - 4 -2-3x or -3x-2 Notice that after the subtraction is applied to the 3x and the 4 in the second line, the parentheses are dropped and there are like terms to be combined to simplify the expression further. The order the terms are written does not matter, as long as the correct sign stays with its corresponding term.

The sequence has the 4th and 5th terms of 33 and -51 respectively. Each term is found by adding the previous 2 terms and multiplying by -3. What is the value of a2? a2, a3, 33, -51...

5 To solve, work backwards using opposite operations. -51 ÷ -3 = 17, so 17 is the sum of 33 and a3. 33 + a3 = 17 so the value of a3 is -16. This process must be repeated using 33 and -16 for an answer of 5.

Which of the following correctly simplifies the left side of the equation below? 3x - 4 + 6x + 5 - 4x - 2 = 16

5x - 1 = 16 This is the only answer option that combines the terms correctly. Another way to write the problem is 3x + 6x - 4x - 4 - 2 + 5 = 16; this can then be written 5x - 1 = 16.

Simplify the expression: 3x + 2(3x - 2) + 5 - 2(2x + 1)

5x-1 First, each value from the front of each set of parentheses must be distributed into the parentheses and multiplied appropriately, with attention to positive and negative signs: 3x + 6x - 4 + 5 - 4x - 2 Next, like terms must be collected to be combined: (3x + 6x - 4x) + (-4 + 5 - 2) Finally, like terms can be combined: 5x - 1

John's car can drive 50 miles per gallon of gas, he wants to take a trip that is 750 miles. Gas currently costs $3.45 per gallon. He drives an average of 60 miles per hour during the trip. Which of the following expressions can solve for how many gallons of gas must he purchase for the trip?

750 \div÷ 50 750 \div÷ 50 represents the 750 miles traveled divided by the 50 miles per gallon his car can travel. The answer would provide the number of gallons needed.

Inequalities: Is Less Than or Equal To

<, all the numbers to the left of that point on the number line, including that point x≤-1

Inequalities: Is Greater Than

>, all the number to the right of that point on the number line x>-1

The formula A=b⋅h

A = area of the rectangle, b = base of the rectangle, h = height of the rectangle.

Variable

A letter or non-numeric symbol that represents an unknown value. In the expression 2x+4y-5, there are 2 variables: x and y.

Combine Like Terms

A method of simplifying an algebraic expression by adding or subtracting the coefficients of like terms. 2x + 4x = 6x

Coefficient

A number that multiplies a variable. In the expression 3x+1, 3 is the coefficient of x.

Constant

A number without a variable. It is called a constant because its value does not change (it stays constant). In the expression 2x+4y-5, there is one constant: -5.

System of Equations

A set of two or more equations or inequalities with the same set of variables, or unknowns. 2x + 4y = 10 5x - 6y = 12

Inequality

A statement that 2 expressions are not equal. 2x + 3 > 6

What is the 10th term of the geometric sequence 50, 40, 32, 25.6, 20.48,...?

A(10) ≈ 6.71 A(10) ≈ 6.71 comes from seeing that the sequence is shown to have a₁ = 50, r = 40 ÷ 50 = 0.8, and n = 10. The question is answered correctly by keeping an appropriate variable expression in the place of A(n) and inputting all known values so that the formula A(n) = a₁(r)ⁿ⁻¹ becomes A(10) = 50(0.8)¹⁰⁻¹. When the order of operations is followed correctly, exponents will be simplified before any multiplication occurs. The resulting equation becomes A(10) = 50(0.8)⁹ and then A(10) = 50(0.134217728) [or, fractions can be used for r so that r = 4/5 and r⁹ = 262,144/1,953,125 so that the equation is A(10) = 50(262,144/1,953,125)]. The final answer is the product of 50 and the decimal or fraction above, and so A(10) ≈ 6.71 when rounded to the nearest hundredth.

Linear Expression

An expression that does not contain any exponents 2x + 3

Distributive Property

An number in front of a group of terms will multiply all terms in the grouping individually a(b+c) = ab + ac

ex; In the x column, the values increase by +1 every time, and in the y column, the values decrease by -2 every time. In this case, the rate of change is

Change of y over change of x -2/1 = -2

Mr. Sudu is a waiter. His total weekly earnings consist of a wage of $6 per hour plus approximately 15% in tips on his total sales for the week. One week Mr. Sudu worked 25 hours and had total sales of z dollars. Which of the following represents his total weekly earning in dollars, E, for that week?

E=0.15z+150

Term

Each part of an expression that is separated by a + or - sign. In the expression 2x+4y-5, there are 3 terms: 2x, 4y, and -5.

What type of sequence is the following number sequence? 4, 12, 20, 28, 36, 44...

It is an arithmetic sequence; you add 8 every time to get the next term. Arithmetic sequences have a common difference. This sequence has a common difference of 8 (12-4=8; 30-12=8; 28-20=8;36-28=8; 44-36=8).

Perpendicular Lines

Lines that intersect at a right (90º) angle. They have slopes that are opposite reciprocals, meaning their signs (positive or negative) are opposite and their fractions are flipped. y=2/5x+3 and y=-5/2x-4

Which situation could best be represented by the equation: 12x = 54?

Marty made car payments on her car for 54 months until it was paid off. What is x, the number of years it took Marty to pay off her car?

Expression

Numbers, symbols, and operators grouped together to show the value of something. 3x - 4y + 6 is an expression. Note that it differs from an equation because there is no equal sign and therefore cannot be solved, only simplified.

When calculating slope, make sure you don't switch the rise and the run. Remember that the rise is the number in the numerator of the slope fraction and should match the vertical movement of the line. The run is in the denominator of the slope fraction and matches the horizontal movement of the line.

Rise = numerator = vertical Run = denominator = horizontal

You can determine the slope (m) by counting the rise and run between two points, and writing them as a ratio

Rise over run

Steps For Graphing an Equation in Slope-Intercept Form.

Start by identifying the slope (m) and the y-intercept (b) from the equation. Begin with b: Plot the y-intercept. Move with m: Starting at the y-intercept, count out the rise and run of the slope and extend the line. Double-check your work: particularly the signs of the slope and the location of the y-intercept.

Like Terms

Terms with the same variable and exponent combination. 2x and 5x

Independent Variable (in functions)

The input into a function, representing the variable that is known. Normally, it is represented by the variable x.

Elona is writing an essay. She writes 3 pages every hour. If her essay is 10 pages long, how many hours did she spend writing? In the problem above, what is the dependent variable?

The number of pages Elona writes. The number of pages she writes depends on how long she spent writing and is, therefore, the dependent variable.

Dependent Variable (in functions)

The output of a function, or the result after solving the function using the independent variable. It commonly represented by the variable y.

In the example with Keisha above, the problem clearly states to write an equation. Consider the slightly different problem below: Keisha had $45. She went to work, where she earned an hourly wage of $13.50 per hour. How many hours does she have to work if she wants to have at least $300 in total?

The problem states that she wants to have "at least $300." This means that the amount of money she has can be exactly $300 or more than $300 → total money 300≥. As previously discussed, the expression that represents her total money is still 45+13.50h, so the inequality that represents this situation is 45+13.50h≥300.

For example, 2x + 3y = 13 and 4x - y = 5.

The solution would be x = 2 and y = 3.

volume of a rectangular prism

V=lwh

The Parent Teacher Organization at Douglass Elementary baked cookies. The ingredients to make each batch of cookies cost $3. Each batch made 20 cookies. The PTO sold each cookie for $0.50. They produced b batches of cookies, and sold every single one of them. What is a valid expression, in terms of b, for the profit that the PTO made for their cookie sale?

[(0.5)(20) -3]b In general, Profit = Revenue - Expenses. The revenue that the PTO brought in from their bake sale was $0.50 for every cookie sold. There were 20 cookies in each of b batches of cookies made and sold. Therefore, there were a total of 20b cookies produced and sold. With each cookie selling for $0.50, the total revenue from the sale was 0.50 × 20b, which can also be expressed as (0.5)(20b) or (0.5)(20)b. The expense to produce the cookies was $3 for every batch. Therefore, expenses = 3b. Profit can now be expressed as the difference between revenue and expenses: (0.5)(20)b - 3b. The answer choices show b factored out, and so the answer [(0.5)(20) -3]b can be selected.

If something is outside the parentheses

apply it to everything inside.

ex; We are given a number of miles, which is a y-value. To determine the number of hours that corresponds to 8 miles, we need to determine the x-value of the ordered pair that falls on the line with a y-value of 8, or (x, 8).

ex; To do this, go to 8 on the y-axis (miles). Draw a horizontal line from 8 on the y-axis to the line. Draw a vertical line down from the line to the x-axis (hours). It intersects at 4 on the x-axis. This gives us the ordered pair (4, 8), as shown on the graph to the right. It takes George 4 hours to go 8 miles. So, it will take his 4 hours to get to his friends house.

Sequences that have a common ratio, or multiplier, are called

geometric sequences

Traditionally, the coefficient is written

in front of the variable

>

is greater than is more than

<

is less than is fewer than

Make sure you match the sign of the slope with the direction of the line. Remember:

positive slopes should increase from left to right (uphill) negative slopes should decrease from left to right (downhill)

Profit =

revenue - cost

Look for the word "per" to clue you into a rate, which is usually

the slope.

What is x in terms of y for the equation: 74 - 6x = 9y

x = (9y -74) ÷ -6 To isolate x, first subtract 74 from each side so the equation is -6x = 9y -74. Then divide both sides by -6: x = (9y -74) ÷ -6

When presented with the phrase "the difference of" or "the sum of," add or subtract the values in the order they are written in the problem. For example: The difference of x and 3 translates to

x-3 NOT 3-x

Five less than x means that you take five away from x:

x−5 NOT 5-x

Which of the following equations is written in slope-intercept form?

y = 3x + 5

The dependent variable (_____) depends on the independent variable (____).

y;x

If a relationship is linear, an equation in slope-intercept form, or

y=mx+b, can be used to represent the relationship.