This exam was adminstered in January 2024 .

### June 2024 Algebra, Part II

Each correct answer is worth up to 2 credits. Partial credit can be given. Work must be shown or explained.

**25.** * Solve 5(x - 2) < 3x + 20 algebraically. *

**Answer: **

Distribute the 5 and use Inverse Operations.

5(x - 2) __<__ 3x + 20

5x - 10 __<__ 3x + 20

5x __<__ 3x + 30

2x __<__ 30

x __<__ 15

This wasn't a "trick" question which required flipping the direction of the inequality symbol because no negatives where multiplied or divided.

**26.** * Given g(x) = x _{3} + 2x^{2} - x, evaluate g(-3). *

**Answer: **

Substitute and evaluate. Remember the order of operations. Or put it into your calculator.

g(-3) = (-3)_{3} + 2(-3)^{2} - (-3) = -6

If you put this into your calculator, remember to use all the parentheses that I used above to avoid a mistake.

You could also graph y = x_{3} + 2x^{2} - x and check the table of values for x = -3.

Whatever you do, write something on the paper other than just the answer, which is only worth one point. I suggest you write down the substition. It will be assumed that you calculated it in your head or with a calculator, so you don't need to write down every step.

**27.** *Given the relation R = {(-1,1), (0,3), (-2,-4), (x,5)}.State a value for x that will make this relation a function.Explain why your answer makes this a function.*

**Answer: **

The relation is a function as long as the x values do not repeat.

This means you can write *any* number **except** -1, 0, or -2.

That is, you can write x = 2, x = 10, x = π, x = 1/2 or pretty much anything you want, so long as it's a number other than the three already in the set.

**28.** *A survey of 150 students was taken. It was determined that 2/3 of the students play video games.Of the students that play video games, 85 also use social media.Of the students that do not play video games, 20% do not use social media.*

*Complete the two-way frequency table *

**Answer: **

Start with the numbers you know and work backward to find the missing information.

The grand total, bottom right corner, is 150. If 2/3 play video games, that means that the bottom row will be divided into 100 and 50.

Of the students who play video games (100 students), 85 use social media, which means that 100 - 85 = 15 do not.

Of the ones not playing video games (50 students), 20% don't use social media. Since .20 * 50 = 10, that means 50 - 10 = 40 do use social media.

Add the rows across. You will get the following table:

**29.** *Use the method of completing the square to determine the exact values of x for the equation x ^{2} + 10x - 30 = 0. *

**Answer: **

They stated that you must complete the square. If you use any other method, then the most you can get is 1 point for a correct value of x.

To complete the square, you need to divide the middle value by 2, which is 10/2 = 5. That means that (x + 5)^{2} will be part of the solution. To complete the square you need to add 25 to each side of the equation.

x^{2} + 10x - 30 = 0

x^{2} + 10x = 30

x^{2} + 10x + 25 = 55

(x + 5)^{2} = 55

x + 5 = __+__√(55)

x = -5 __+__√(55)

**30.** * Factor 20x ^{3} - 45x completely.*

**Answer: **

Whenever you see "Factor completely", you can be pretty sure that there will be more than one step. Show them all.

First, look for common factors in each of the terms. Then factor what remains. There are two terms separated by a minus sign, so keep an eye out for a Difference of Squares.

20x^{3} - 45x = 5x (4x^{2} - 9) = 5x (2x + 3)(2x - 3)

Note that this is an expression and not an equation. Do NOT attempt to "solve" it, or you will lose a point.

**End of Part II**

How did you do?

Questions, comments and corrections welcome.

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