Solve the Absolute Value Inequality: |2x - 5| > 7 Step by Step

Q: Why do I need to solve two separate inequalities?

Absolute value represents distance from zero. For |A| > 7, the expression A can be either greater than 7 OR less than -7 to have a distance greater than 7 from zero.

Q: How do I remember which direction the inequality signs go?

Think of it this way: if |A| > B, then A is either way to the right (A > B) or way to the left (A

Q: What's the difference between > and < in absolute value inequalities?

For |A| > B, you get two separate regions (A > B or A one combined region (-B

Q: How can I check if my final answer is correct?

Pick test values from each solution region! For $ x 6 $, try x = -2 and x = 7. Both should make the original inequality true.

Q: Why is the answer written as 'or' instead of 'and'?

The solution represents all x-values that work. Since x can be in either region (less than -1 OR greater than 6), we use 'or' to show both possibilities.

Absolute Value Inequalities with Greater Than

Solve the inequality:

$|2x - 5| > 7$

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve the inequality:

$|2x - 5| > 7$

Step-by-step solution

To solve $|2x - 5| > 7$ , we consider the definition of absolute value inequality $|A| > B$ , which means $A > B$ or $A < -B$ .

Thus, $2x - 5 > 7$ or $2x - 5 < -7$ .

Let's solve these inequalities separately:

1. $2x - 5 > 7$

Add 5 to both sides:

$2x > 12$

Divide by 2:

$x > 6$

2. $2x - 5 < -7$

Add 5 to both sides:

$2x < -2$

Divide by 2:

$x < -1$

Therefore, the solution is $x < -1 \text{ or } x > 6$ .

Final Answer

$x < -1 \text{ or } x > 6$

Key Points to Remember

Essential concepts to master this topic

Rule: |A| > B means A > B or A < -B
Technique: Solve 2x - 5 > 7 and 2x - 5 < -7 separately
Check: Test x = -2: |2(-2) - 5| = |-9| = 9 > 7 ✓

Common Mistakes

Avoid these frequent errors

Solving |2x - 5| > 7 as one inequality
Don't solve 2x - 5 > 7 only = missing half the solution! This ignores when the expression inside is negative. Always split into two cases: A > B OR A < -B.

Practice Quiz

Test your knowledge with interactive questions

Given:

$\left|2x-1\right|>-10$

Which of the following statements is necessarily true?

FAQ

Everything you need to know about this question

Why do I need to solve two separate inequalities?

Absolute value represents distance from zero. For |A| > 7, the expression A can be either greater than 7 OR less than -7 to have a distance greater than 7 from zero.

How do I remember which direction the inequality signs go?

Think of it this way: if |A| > B, then A is either way to the right (A > B) or way to the left (A < -B) on the number line.

What's the difference between > and < in absolute value inequalities?

For |A| > B, you get two separate regions (A > B or A < -B). For |A| < B, you get one combined region (-B < A < B).

How can I check if my final answer is correct?

Pick test values from each solution region! For $x < -1 \text{ or } x > 6$ , try x = -2 and x = 7. Both should make the original inequality true.

Why is the answer written as 'or' instead of 'and'?

The solution represents all x-values that work. Since x can be in either region (less than -1 OR greater than 6), we use 'or' to show both possibilities.

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