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

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Example Questions

Example Question #5 : Inequalities

Solve |x – 5| ≤ 1

Possible Answers:

4 ≤ x ≤ 6

None of the answers are correct

-1 ≤ x ≤ 1

x ≤ 4 or x ≥ 6

0 ≤ x ≤ 1

Correct answer:

4 ≤ x ≤ 6

Explanation:

Absolute values have two answers: a positive one and a negative one. Therefore,

-1 ≤ x – 5≤ 1 and solve by adding 5 to all sides to get 4 ≤ x ≤ 6.

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Example Question #1 : Inequalities

Solve $|z - 3| \leq 5$

Possible Answers:

No solutions

All real numbers

$-2 \geq z$

$z \leq 8$

$-2 \leq z \leq 8$

Correct answer:

$-2 \leq z \leq 8$

Explanation:

Absolute value is the distance from the origin and is always positive.

So we need to solve $z - 3 \leq 5$ and $z - 3 \geq -5$ which becomes a bounded solution.

Adding 3 to both sides of the inequality gives $z \leq 8$ and $z \geq -2$ or in simplified form $-2 \leq z\leq 8$

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

Given the inequality $\left | 12-2x\right |>2,$ which of the following is correct?

Possible Answers:

x<5 or x>7

x<-5 or x>7

-5<x<7

x<-7 or x>5

5<x<7

Correct answer:

x<5 or x>7

Explanation:

First separate the inequality $\left | 12-2x\right |>2$ into two equations.

12-2x>2

12-2x<-2

Solve the first inequality.

12-2x>2

-2x>-10

x<5

Solve the second inequality.

12-2x<-2

-2x<-14

x>7

Thus, x<5 or x>7 .

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

What values of x make the following statement true?

|x – 3| < 9

Possible Answers:

–12 < x < 6

6 < x < 12

–6 < x < 12

x < 12

–3 < x < 9

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 #2 : How To Find The Solution To An Inequality With Addition

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

Possible Answers:

|w|

w²

|w|^0.5

3w/2

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 #3 : How To Find The Solution To An Inequality With Addition

Solve for .

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

Possible Answers:

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

$z\geq 8$

$-3\leq z\leq 5$

$-2\leq z\leq 8$

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

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 #2 : How To Find The Solution To An Inequality With Addition

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

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

Possible Answers:

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

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

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

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

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

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

Simplify the following inequality

$15 \le 2x + 7 \le 21$ .

Possible Answers:

$7 \le x \le 11$

$6 \le x \le 12$

$4 \le x \le 7$

$8 \le x \le 14$

$-11 \le x \le -7$

Correct answer:

$4 \le x \le 7$

Explanation:

For a combined inequality like this, you just need to be careful to perform your operations on all the parts of the inequality. Thus, begin by subtracting from each member:

$8\le 2x \le14$

Next, divide all of the members by :

$4 \le x \le 7$

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Example Question #7 : How To Find The Solution To An Inequality With Addition

Simplify

3x + 15 < 42 .

Possible Answers:

x > -9

x > -5

x < 27

x < 25

x < 9

Correct answer:

x < 9

Explanation:

Simplifying an inequality like this is very simple. You merely need to treat it like an equation—just don't forget to keep the inequality sign.

First, subtract from both sides:

3x < 27

Then, divide by :

x < 9

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

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #5 : Inequalities

Example Question #1 : Inequalities

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

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

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

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

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

Example Question #1 : 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 #7 : How To Find The Solution To An Inequality With Addition

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