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

## Example Questions

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### Example Question #1 : Inequalities

Solve |x – 5| ≤ 1

0 ≤ x ≤ 1

None of the answers are correct

-1 ≤ x ≤ 1

4 ≤ x ≤ 6

x ≤ 4 or x ≥ 6

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.

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

Solve

All real numbers

No solutions

Explanation:

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

So we need to solve  and  which becomes a bounded solution.

Adding 3 to both sides of the inequality gives  and  or in simplified form

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

Given the inequality  which of the following is correct?

or

or

or

or

Explanation:

First separate the inequality   into two equations.

Solve the first inequality.

Solve the second inequality.

Thus,   or .

### 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

x < 12

–3 < x < 9

–12 < x < 6

6 < x < 12

–6 < x < 12

–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.

### Example Question #11 : Inequalities

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

w/2

|w|

3w/2

|w|0.5

w2

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.

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

Solve for .

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.

### Example Question #231 : Algebra

If and , then which of the following could be the value of ?

Explanation:

To solve this problem, add the two equations together:

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

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

What values of  make the statement  true?

Explanation:

First, solve the inequality :

Since we are dealing with absolute value,  must also be true; therefore:

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

Simplify the following inequality

.

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:

Next, divide all of the members by :

Simplify

.