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SAT Math : How to find the solution to an inequality with addition

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Example Questions

Example Question #1921 : Act Math

What values of x make the following statement true?

|x – 3| < 9

Possible Answers:

–12 < x < 6

6 < x < 12

–6 < x < 12

–3 < x < 9

x < 12

Correct answer:

–6 < x < 12

Explanation:

Solve the inequality by adding 3 to both sides to get x < 12. Since it is absolute value, x – 3 > –9 must also be solved by adding 3 to both sides so: x > –6 so combined.

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Example Question #1922 : Act Math

If –1 < w < 1, all of the following must also be greater than –1 and less than 1 EXCEPT for which choice?

Possible Answers:

3w/2

w²

|w|^0.5

|w|

w/2

Correct answer:

3w/2

Explanation:

3w/2 will become greater than 1 as soon as w is greater than two thirds. It will likewise become less than –1 as soon as w is less than negative two thirds. All the other options always return values between –1 and 1.

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Example Question #1923 : Act Math

Solve for .

$\left | z-3 \right |\geq 5$

Possible Answers:

$-3\leq z\leq 5$

$z\geq 8$

$z\leq -2\ \text{or}\ z\geq 8$

$z\leq -3\ \text{or}\ z\geq 5$

$-2\leq z\leq 8$

Correct answer:

$z\leq -2\ \text{or}\ z\geq 8$

Explanation:

Absolute value problems always have two sides: one positive and one negative.

First, take the problem as is and drop the absolute value signs for the positive side: z – 3 ≥ 5. When the original inequality is multiplied by –1 we get z – 3 ≤ –5.

Solve each inequality separately to get z ≤ –2 or z ≥ 8 (the inequality sign flips when multiplying or dividing by a negative number).

We can verify the solution by substituting in 0 for z to see if we get a true or false statement. Since –3 ≥ 5 is always false we know we want the two outside inequalities, rather than their intersection.

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Example Question #1925 : Act Math

What values of make the statement $\left |5x-9 \right |\geq6$ true?

Possible Answers:

$x\geq4,x\leq -\frac{1}{2}$

$x\geq3, x\leq \frac{3}{5}$

$x\geq15,x\leq \frac{2}{5}$

$x\geq5,x\leq \frac{1}{5}$

$x\geq6,x\leq \frac{1}{3}$

Correct answer:

$x\geq3, x\leq \frac{3}{5}$

Explanation:

First, solve the inequality $5x-9 \geq6$ :

$5x-9 \geq6$

$5x\geq15$

$x\geq3$

Since we are dealing with absolute value, $5x-9\leq-6$ must also be true; therefore:

$5x-9\leq-6$

$5x\leq3$

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

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

Solve: $3x+2\le 49$

Possible Answers:

$x\le \frac{17}{3}$

$x\le \frac{47}{3}$

$x\ge \frac{47}{3}$

$x\le 17$

$x\le 47$

Correct answer:

$x\le \frac{47}{3}$

Explanation:

To solve $3x+2\le 49$ , isolate .

$3x\le 47$

Divide by three on both sides.

$x\le \frac{47}{3}$

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Example Question #21 : Inequalities

Solve for .

-5x+17>62

Possible Answers:

x>-9

x=-9

$x\leq-9$

$x\geq-9$

x<-9

Correct answer:

x<-9

Explanation:

We want to isolate the variable on one side and numbers on another side. Treat like a normal equation.

-5x+17>62 Subtract on both sides.

-5x>45 Divide on both sides. Remember to flip the sign.

x<-9

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

Solve for .

4x+7<5x+13

Possible Answers:

x>-6

$x\geq-6$

x=-6

$x\leq-6$

x<-6

Correct answer:

x>-6

Explanation:

We want to isolate the variable on one side and numbers on another side. Treat like a normal equation.

4x+7<5x+13 Subtract 4x,13 on both sides.

x<-6

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Example Question #6 : How To Find The Solution To An Inequality With Addition

Solve for .

$\left | x+5\right |<8$

Possible Answers:

x=3, x=-13

x>3, x<-13

x>3

-13<x<3

x<-13

Correct answer:

-13<x<3

Explanation:

We want to isolate the variable on one side and numbers on another side. Treat like a normal equation.

$\left | x+5\right |<8$ We need to set-up two equations since its absolute value.

x+5<8 Subtract on both sides. x<3

-(x+5)<8 Divide on both sides which flips the sign.

x+5>-8 Subtract on both sides. x>-13

Since we have the 's being either greater than or less than the values, we can combine them to get -13<x<3 .

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Example Question #23 : Inequalities

Solve for .

$\left | x+7\right |\leq3x+13$

Possible Answers:

$-3\leq x$

$3\leq x\leq 5$

$x\leq -3, x\leq -5$

$-5\leq x\leq -3$

$-5\leq x$

Correct answer:

$-3\leq x$

Explanation:

We want to isolate the variable on one side and numbers on another side. Treat like a normal equation.

$\left | x+7\right |\leq3x+13$ We need to set-up two equations since it's absolute value.

$x+7\leq 3x+13$ Subtract x, 7 on both sides.

$-6\leq 2x$ Divide on both sides.

$-3\leq x$

$-(x+7)\leq 3x+13$ Distribute the negative sign to each term in the parenthesis.

$-x-7\leq3x+13$ Add and subtract on both sides.

$-20\leq4x$ Divide on both sides.

$-5\leq x$ We must check each answer. Let's try .

$\left | 0+7\right |<3(0)+13$ 7<13 This is true therefore $-3\leq x$ is a correct answer. Let's next try .

$\left | -4+7\right |<3(-4)+13$ 3<1 This is not true therefore $-5\leq x$ is not correct.

Final answer is just $-3\leq x$ .

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Example Question #1924 : Act Math

If $x+1< 4$ and $y-2<-1$ , then which of the following could be the value of x+y ?

Possible Answers:

Correct answer:

Explanation:

To solve this problem, add the two equations together:

$x+1<4$

$y-2<-1$

$x+1+y-2<4-1$

$x+y-1<3$

$x+y<4$

The only answer choice that satisfies this equation is 0, because 0 is less than 4.

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SAT Math : How to find the solution to an inequality with addition

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #1921 : Act Math

Example Question #1922 : Act Math

Example Question #1923 : Act Math

Example Question #1925 : Act Math

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

Example Question #21 : Inequalities

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

Example Question #6 : How To Find The Solution To An Inequality With Addition

Example Question #23 : Inequalities

Example Question #1924 : Act Math

All SAT Math Resources

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