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SAT Math : Inequalities

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Example Question #7 : Inequalities

What is the solution set of the inequality $\dpi{100} \small 3x+8<35$ ?

Possible Answers:

$\dpi{100} \small x<35$

$\dpi{100} \small x<27$

$\dpi{100} \small x<9$

$\dpi{100} \small x>27$

$\dpi{100} \small x>9$

Correct answer:

$\dpi{100} \small x<9$

Explanation:

We simplify this inequality similarly to how we would simplify an equation

$\dpi{100} \small 3x+8-8<35-8$

$\dpi{100} \small \frac{3x}{3}<\frac{27}{3}$

Thus $\dpi{100} \small x<9$

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Example Question #8 : Inequalities

What is a solution set of the inequality 2x+12>42 ?

Possible Answers:

x>4

x<15

$x>\frac{3}{2}$

x<9

x>15

Correct answer:

x>15

Explanation:

In order to find the solution set, we solve 2x+12>42 as we would an equation:

2x+12>42

2x>30

x>15

Therefore, the solution set is any value of x>15 .

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Example Question #1 : How To Find The Solution To An Inequality With Division

Which of the following could be a value of , given the following inequality?

$8-7x\leq11x+4$

Possible Answers:

$-\frac{1}{3}$

$\frac{1}{9}$

$-\frac{2}{3}$

$\frac{4}{9}$

$-\frac{2}{9}$

Correct answer:

$\frac{4}{9}$

Explanation:

The inequality that is presented in the problem is:

$8-7x\leq11x+4$

Start by moving your variables to one side of the inequality and all other numbers to the other side:

$-7x-11x\leq4-8$

$-18x\leq-4$

Divide both sides of the equation by . Remember to flip the direction of the inequality's sign since you are dividing by a negative number!

$x\geq\frac{-4}{-18}$

Reduce:

$x\geq\frac{2}{9}$

The only answer choice with a value greater than $\frac{2}{9}$ is $\frac{4}{9}$ .

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Example Question #1 : How To Find The Solution To An Inequality With Division

If $2\leq n \leq12$ and $9\leq p \leq10$ , which of the following gives the set of possible values of $\frac{n}{p}$ ?

Possible Answers:

$\frac{2}{9} \leq \frac{n}{p} \leq \frac{6}{5}$

$\frac{1}{5} \leq \frac{n}{p} \leq \frac{6}{5}$

$\frac{1}{5}\leq \frac{n}{p} \leq \frac{4}{3}$

$0 \leq \frac{n}{p} \leq \frac{1}{5}$

$\frac{2}{9} \leq \frac{n}{p} \leq \frac{5}{6}$

Correct answer:

$\frac{1}{5}\leq \frac{n}{p} \leq \frac{4}{3}$

Explanation:

To get the lowest value, you need the lowest numerator and the highest denominator. That would be $\frac{2}{10}$ or reduced to be $\frac{1}{5}$ . For the highest value, you need the highest numerator and the lowest denominator. That would be $\frac{12}{9}$ or $\frac{4}{3}$ .

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

Give the solution set of this inequality:

$|- 5 x + 32| \le 38$

Possible Answers:

$( -\infty, -1.2] \cup [14 , \infty)$

$( -\infty, -14] \cup [1.2, \infty)$

The inequality has no solution.

[-1.2, 14 ]

[ -14 , 1.2]

Correct answer:

[-1.2, 14 ]

Explanation:

The inequality $|- 5 x + 32| \le 38$ can be rewritten as the three-part inequality

$-38 \le - 5 x + 32 \le 38$

Isolate the in the middle expression by performing the same operations in all three expressions. Subtract 32 from each expression:

$-38 - 32 \le - 5 x + 32 - 32 \le 38 - 32$

$-70 \le - 5 x \le 6$

Divide each expression by , switching the direction of the inequality symbols:

$-70\div (-5) \ge - 5 x\div (-5) \ge 6 \div (-5)$

$14 \ge x \ge -1.2$

This can be rewritten in interval notation as [-1.2, 14 ] .

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

Give the solution set of this inequality:

- 12 - | -4y + 52 | > 32

Possible Answers:

$(-\infty, -24 ]\cup [2, \infty)$

The inequality has no solution.

[-24,2 ]

$(-\infty, -2 ]\cup [24, \infty)$

[-2,24 ]

Correct answer:

The inequality has no solution.

Explanation:

In an absolute value inequality, the absolute value expression must be isolated first, as follows:

- 12 - | -4y + 52 | > 32

Adding 12 to both sides:

- 12 - | -4y + 52 | + 12 >32+ 12

- | -4y + 52 | >44

Multiplying both sides by , and switching the inequality symbol due to multiplication by a negative number:

$- | -4y + 52 | \cdot (-1 ) < 44 \cdot (-1 )$

| -4y + 52 | < -44

We do not need to go further. An absolute value expression must always be greater than or equal to 0; it is impossible for the expression | -4y + 52 | to be less than any negative number. The inequality has no solution.

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SAT Math : Inequalities

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #7 : Inequalities

Example Question #8 : Inequalities

Example Question #1 : How To Find The Solution To An Inequality With Division

Example Question #1 : How To Find The Solution To An Inequality With Division

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

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

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