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PSAT Math : PSAT Mathematics

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Example Questions

Example Question #22 : Inequalities

Solve for x

$\small 3x+7 \geq -2x+4$

Possible Answers:

$\small x \leq -\frac{3}{5}$

$\small x \geq -\frac{3}{5}$

$\small x \geq \frac{3}{5}$

$\small x \leq \frac{3}{5}$

Correct answer:

$\small x \geq -\frac{3}{5}$

Explanation:

$\small 3x+7 \geq -2x+4$

$\small 3x \geq -2x-3$

$\small 5x \geq -3$

$\small x\geq -\frac{3}{5}$

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Example Question #4 : How To Find The Solution To An Inequality With Multiplication

We have x^2-4<0 , find the solution set for this inequality.

Possible Answers:

x=0

-2<x<2

x>-2

x>2, x<-2

x<2

Correct answer:

-2<x<2

Explanation:

$x^2-4<0 \Rightarrow x^2<4 \Rightarrow -2<x<2$

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Example Question #23 : Inequalities

Fill in the circle with either $<$ , $>$ , or $=$ symbols:

$(x-3)\circ\frac{x^2-9}{x+3}$ for $x\geq 3$ .

Possible Answers:

None of the other answers are correct.

The rational expression is undefined.

$(x-3)< \frac{x^2-9}{x+3}$

$(x-3)> \frac{x^2-9}{x+3}$

$(x-3)=\frac{x^2-9}{x+3}$

Correct answer:

$(x-3)=\frac{x^2-9}{x+3}$

Explanation:

$(x-3)\circ\frac{x^2-9}{x+3}$

Let us simplify the second expression. We know that:

$(x^2-9)=(x+3)(x-3)$

So we can cancel out as follows:

$\frac{x^2-9}{x+3}=\frac{(x+3)(x-3)}{(x+3)}=x-3$

$(x-3)=\frac{x^2-9}{x+3}$

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Example Question #21 : Inequalities

What is the greatest value of that makes

$-45 \leq \frac{x - 7}{- 6} \leq - 22$

a true statement?

Possible Answers:

228

277

Correct answer:

277

Explanation:

Find the solution set of the three-part inequality as follows:

$-45 \leq \frac{x - 7}{- 6} \leq - 22$

$-45 \cdot (-6) \geq \frac{x - 7}{- 6} \cdot (-6) \geq - 22 \cdot (-6)$

$270 \geq x-7 \geq 132$

$270 + 7 \geq x-7 + 7 \geq 132 + 7$

$27 7 \geq x \geq 139$

The greatest possible value of is the upper bound of the solution set, which is 277.

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Example Question #5 : How To Find The Solution To An Inequality With Multiplication

What is the least value of that makes

$-45 \leq \frac{x - 7}{- 6} \leq - 22$

a true statement?

Possible Answers:

139

125

Correct answer:

139

Explanation:

Find the solution set of the three-part inequality as follows:

$-45 \leq \frac{x - 7}{- 6} \leq - 22$

$-45 \cdot (-6) \geq \frac{x - 7}{- 6} \cdot (-6) \geq - 22 \cdot (-6)$

$270 \geq x-7 \geq 132$

$270 + 7 \geq x-7 + 7 \geq 132 + 7$

$27 7 \geq x \geq 139$

The least possible value of is the lower bound of the solution set, which is 139.

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Example Question #2 : How To Find The Solution To An Inequality With Multiplication

Give the solution set of the inequality:

$-7 \leq - \frac{3}{4}x < -4$

Possible Answers:

$\left [- 9\frac{1}{3},- 5\frac{1}{3} \right )$

$\left ( 5\frac{1}{3}, 9 \frac{1}{3} \right ]$

$\left (3, 5 \frac{1}{4} \right ]$

$\left [ - 5\frac{1}{4}, -3 \right )$

None of the other responses gives the correct answer.

Correct answer:

$\left ( 5\frac{1}{3}, 9 \frac{1}{3} \right ]$

Explanation:

Divide each of the three expressions by $-\frac{3}{4}$ , or, equivalently, multiply each by its reciprocal, $-\frac{4}{3}$ :

$-7 \leq - \frac{3}{4}x < -4$

$-7 \cdot \left ( - \frac{4}{3} \right ) \geq - \frac{3}{4}x \cdot \left ( - \frac{4}{3} \right ) > -4 \cdot \left ( - \frac{4}{3} \right )$

$\frac{28}{3} \geq x > \frac{16}{3}$

$5 \frac{1}{3} < x \leq 9 \frac{1}{3}$

or, in interval form,

$\left ( 5\frac{1}{3}, 9 \frac{1}{3} \right ]$ .

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Example Question #8 : How To Find The Solution To An Inequality With Multiplication

Give the solution set of the following inequality:

$32 < \frac{x - 4}{5} < 42$

Possible Answers:

(133,173)

(180, 230)

(164, 214)

None of the other responses gives the correct answer.

(148, 188)

Correct answer:

(164, 214)

Explanation:

$32 < \frac{x - 4}{5} < 42$

$32\cdot 5 < \frac{x - 4}{5} \cdot 5< 42 \cdot 5$

160 < x - 4 < 210

160+ 4< x - 4 + 4< 210 + 4

164 < x < 214

or, in interval notation, (164, 214) .

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Example Question #3 : How To Find The Solution To An Inequality With Multiplication

Which of the following numbers could be a solution to the inequality 3x<x<-2x ?

Possible Answers:

$\frac{1}{2}$

Correct answer:

Explanation:

In order for a negative multiple to be greater than a number and a positive multiple to be less than that number, that number must be negative itself. -4 is the only negative number available, and thus the correct answer.

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Example Question #5 : How To Find The Solution To An Inequality With Division

Each of the following is equivalent to

xy/z * (5(x + y)) EXCEPT:

Possible Answers:

xy(5x + 5y)/z

5x² + y²/z

xy(5y + 5x)/z

5x²y + 5xy²/z

Correct answer:

5x² + y²/z

Explanation:

Choice a is equivalent because we can say that technically we are multiplying two fractions together: (xy)/z and (5(x + y))/1. We multiply the numerators together and the denominators together and end up with xy (5x + 5y)/z. xy (5y + 5x)/z is also equivalent because it is only simplifying what is inside the parentheses and switching the order- the commutative property tells us this is still the same expression. 5x²y + 5xy²/z is equivalent as it is just a simplified version when the numerators are multiplied out. Choice 5x² + y²/z is not equivalent because it does not account for all the variables that were in the given expression and it does not use FOIL correctly.

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Example Question #151 : Algebra

Let S be the set of numbers that contains all of values of x such that 2x + 4 < 8. Let T contain all of the values of x such that -2x +3 < 8. What is the sum of all of the integer values that belong to the intersection of S and T?

Possible Answers:

-2

-7

-3

Correct answer:

-2

Explanation:

First, we need to find all of the values that are in the set S, and then we need to find the values in T. Once we do this, we must find the numbers in the intersection of S and T, which means we must find the values contained in BOTH sets S and T.

S contains all of the values of x such that 2x + 4 < 8. We need to solve this inequality.

2x + 4 < 8

Subtract 4 from both sides.

2x < 4

Divide by 2.

x < 2

Thus, S contains all of the values of x that are less than (but not equal to) 2.

Now, we need to do the same thing to find the values contained in T.

-2x + 3 < 8

Subtract 3 from both sides.

-2x < 5

Divide both sides by -2. Remember, when multiplying or dividing an inequality by a negative number, we must switch the sign.

x > -5/2

Therefore, T contains all of the values of x that are greater than -5/2, or -2.5.

Next, we must find the values that are contained in both S and T. In order to be in both sets, these numbers must be less than 2, but also greater than -2.5. Thus, the intersection of S and T consists of all numbers between -2.5 and 2.

The question asks us to find the sum of the integers in the intersection of S and T. This means we must find all of the integers between -2.5 and 2.

The integers between -2.5 and 2 are the following: -2, -1, 0, and 1. We cannot include 2, because the values in S are LESS than but not equal to 2.

Lastly, we add up the values -2, -1, 0, and 1. The sum of these is -2.

The answer is -2.

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PSAT Math : PSAT Mathematics

Study concepts, example questions & explanations for PSAT Math

All PSAT Math Resources

Example Questions

Example Question #22 : Inequalities

Example Question #4 : How To Find The Solution To An Inequality With Multiplication

Example Question #23 : Inequalities

Example Question #21 : Inequalities

Example Question #5 : How To Find The Solution To An Inequality With Multiplication

Example Question #2 : How To Find The Solution To An Inequality With Multiplication

Example Question #8 : How To Find The Solution To An Inequality With Multiplication

Example Question #3 : How To Find The Solution To An Inequality With Multiplication

Example Question #5 : How To Find The Solution To An Inequality With Division

Example Question #151 : Algebra

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