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

Study concepts, example questions & explanations for ACT Math

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Example Questions

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Example Question #6 : Inequalities

Solve |x – 5| ≤ 1

Possible Answers:

None of the answers are correct

-1 ≤ x ≤ 1

4 ≤ x ≤ 6

0 ≤ x ≤ 1

x ≤ 4 or x ≥ 6

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.

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

Solve

Possible Answers:

No solutions

All real numbers

Correct answer:

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 #7 : Inequalities

Given the inequality  which of the following is correct?

Possible Answers:

 or 

 or 

 or 

Correct answer:

 or 

Explanation:

First separate the inequality   into two equations.

 

Solve the first inequality. 

 

 

Solve the second inequality.

 

Thus,   or .

Example Question #5 : Inequalities

What values of x make the following statement true?

|x – 3| < 9

Possible Answers:

–3 < x < 9

6 < x < 12

–12 < x < 6

x < 12

–6 < 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.

Example Question #27 : Inequalities

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.

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

Solve for .

Possible Answers:

Correct answer:

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

If x+1< 4 and y-2<-1 , then what is the value of ?

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.

Example Question #232 : Gre Quantitative Reasoning

What values of  make the statement  true?

Possible Answers:

Correct answer:

Explanation:

First, solve the inequality :

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

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

Simplify the following inequality

.

Possible Answers:

Correct answer:

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 :

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

Simplify

.

Possible Answers:

Correct answer:

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:

Then, divide by :

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