Solve the Absolute Value Inequality: |x+4| > 13

Q: Why do we use OR instead of AND for the solution?

Think of it this way: the absolute value $ |x+4| $ represents distance. We need values where this distance is greater than 13. This happens when either x is far to the right (x > 9) or far to the left (x

Q: How do I remember when to flip the inequality sign?

The inequality sign only flips when you multiply or divide by a negative number. In this problem, we're just adding/subtracting, so the signs stay the same!

Q: What's the difference between |x+4| > 13 and |x+4| < 13?

Great question! $ |x+4| > 13 $ gives you values outside the interval (uses OR), while \( |x+4| gives values inside the interval (uses AND).

Q: Can I solve this by graphing instead?

Absolutely! Graph $ y = |x+4| $ and the horizontal line $ y = 13 $. The solution is where the V-shaped graph is above the horizontal line.

Q: How do I check if x = -20 works?

Substitute: $ |-20+4| = |-16| = 16 $. Since 16 > 13, yes it works! Any value less than -17 will satisfy the inequality.

Q: What if the absolute value equals exactly 13?

If $ |x+4| = 13 $, then x = 9 or x = -17. But our inequality uses > (not ≥), so these boundary values are not included in the solution.

Absolute Value Inequalities with OR Logic

Given:

$\left|x+4\right|>13$

Which of the following statements is necessarily true?

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Given:

$\left|x+4\right|>13$

Which of the following statements is necessarily true?

Step-by-step solution

To solve the inequality $\left|x + 4\right| > 13$ , we use the property of absolute values, which says that for $\left|a\right| > b$ , it implies $a > b$ or $a < -b$ .

Applying this to our problem, we have:

$x + 4 > 13$ or $x + 4 < -13$ .

Now, let's solve each inequality separately:

First inequality: $x + 4 > 13$

Subtract 4 from both sides to isolate $x$ :

$x > 13 - 4$

$x > 9$

Second inequality: $x + 4 < -13$

Subtract 4 from both sides to isolate $x$ :

$x < -13 - 4$

$x < -17$

Therefore, the solution to the inequality $\left|x + 4\right| > 13$ is $x > 9$ or $x < -17$ .

The correct answer choice is:

$x > 9$ or $x < -17$ .

Final Answer

$x>9$ or $x<-17$

Key Points to Remember

Essential concepts to master this topic

Rule: For |a| > b, split into: a > b OR a < -b
Technique: From x + 4 > 13, subtract 4: x > 9
Check: Test x = 10: |10+4| = 14 > 13 ✓

Common Mistakes

Avoid these frequent errors

Using AND instead of OR when combining solutions
Don't write x > 9 AND x < -17 = impossible solution! No number can be both greater than 9 and less than -17 at the same time. Always use OR for absolute value inequalities with > or ≥.

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 we use OR instead of AND for the solution?

Think of it this way: the absolute value $|x+4|$ represents distance. We need values where this distance is greater than 13. This happens when either x is far to the right (x > 9) or far to the left (x < -17).

How do I remember when to flip the inequality sign?

The inequality sign only flips when you multiply or divide by a negative number. In this problem, we're just adding/subtracting, so the signs stay the same!

What's the difference between |x+4| > 13 and |x+4| < 13?

Great question! $|x+4| > 13$ gives you values outside the interval (uses OR), while $|x+4| < 13$ gives values inside the interval (uses AND).

Can I solve this by graphing instead?

Absolutely! Graph $y = |x+4|$ and the horizontal line $y = 13$ . The solution is where the V-shaped graph is above the horizontal line.

How do I check if x = -20 works?

Substitute: $|-20+4| = |-16| = 16$ . Since 16 > 13, yes it works! Any value less than -17 will satisfy the inequality.

What if the absolute value equals exactly 13?

If $|x+4| = 13$ , then x = 9 or x = -17. But our inequality uses > (not ≥), so these boundary values are not included in the solution.

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Absolute Value and Inequality questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime

Solve the Absolute Value Inequality: |x+4| > 13

Absolute Value Inequalities with OR Logic

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why do we use OR instead of AND for the solution?

How do I remember when to flip the inequality sign?

What's the difference between |x+4| > 13 and |x+4| < 13?

Can I solve this by graphing instead?

How do I check if x = -20 works?

What if the absolute value equals exactly 13?

🌟 Unlock Your Math Potential

More Questions

Inequalities with Absolute Values

Solve |x-4| < 8: Absolute Value Inequality Analysis

Solve |x+2| < 3: Absolute Value Inequality Analysis

Absolute Value Inequality: Determining Outcomes for |x-5| > 11

Understanding Absolute Value Inequality: |x-5| > -11