Solve |4x+12| > 16: Absolute Value Inequality Challenge

Q: Why do we get two separate inequalities from one absolute value inequality?

The absolute value $ |4x+12| $ represents distance from zero. For this distance to be greater than 16, the expression inside can be either very positive (> 16) or very negative (

Q: How do I know when to use 'or' versus 'and' in my final answer?

For absolute value inequalities like $ |A| > B $, always use OR because we want values that satisfy either condition. Use 'and' only when both conditions must be true simultaneously.

Q: What does the solution x > 1 or x < -7 look like on a number line?

Draw two separate regions: everything to the right of 1 (not including 1) and everything to the left of -7 (not including -7). The middle section from -7 to 1 is not part of the solution.

Q: Why isn't x = 1 or x = -7 included in the solution?

Because our inequality uses > (greater than), not ≥. At x = 1: $ |4(1)+12| = 16 $, which equals 16 but doesn't satisfy > 16.

Q: How can I check if my solution regions are correct?

Pick a test point from each region (like x = 2 and x = -8)Substitute into the original inequalityBoth should make $ |4x+12| > 16 $ truePick a point between the regions (like x = 0) - it should make the inequality false

Absolute Value Inequalities with Union Solutions

Given:

$\left|4x+12\right|>16$

Which of the following statements is necessarily true?

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Step-by-step written solution

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Understand the problem

Given:

$\left|4x+12\right|>16$

Which of the following statements is necessarily true?

Step-by-step solution

To solve this inequality, we use the principle that if $|A| > B$ , then $A > B$ or $A < -B$ . Here's a detailed step-by-step solution:

Set up the two inequalities from the absolute value expression:
1. $4x + 12 > 16$
2. $4x + 12 < -16$
For the first inequality, $4x + 12 > 16$ :
- Subtract 12 from both sides to isolate the term with $x$ :
$4x > 16 - 12$
$4x > 4$
- Divide both sides by 4:
$x > 1$
For the second inequality, $4x + 12 < -16$ :
- Subtract 12 from both sides:
$4x < -16 - 12$
$4x < -28$
- Divide both sides by 4:
$x < -7$
The solution is the union of both results:
$x > 1$ or $x < -7$

Therefore, the correct answer among the given choices is $x > 1$ or $x < -7$ .

Final Answer

$x>1$ or $x<-7$

Key Points to Remember

Essential concepts to master this topic

Rule: If |A| > B, then A > B or A < -B
Technique: Solve 4x + 12 > 16 and 4x + 12 < -16 separately
Check: Test x = 2: |4(2) + 12| = |20| = 20 > 16 ✓

Common Mistakes

Avoid these frequent errors

Using 'and' instead of 'or' to connect solutions
Don't write x > 1 AND x < -7 = impossible condition! No number can be both greater than 1 and less than -7 simultaneously. Always use OR to connect the two solution regions for absolute value inequalities.

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 get two separate inequalities from one absolute value inequality?

The absolute value $|4x+12|$ represents distance from zero. For this distance to be greater than 16, the expression inside can be either very positive (> 16) or very negative (< -16).

How do I know when to use 'or' versus 'and' in my final answer?

For absolute value inequalities like $|A| > B$ , always use OR because we want values that satisfy either condition. Use 'and' only when both conditions must be true simultaneously.

What does the solution x > 1 or x < -7 look like on a number line?

Draw two separate regions: everything to the right of 1 (not including 1) and everything to the left of -7 (not including -7). The middle section from -7 to 1 is not part of the solution.

Why isn't x = 1 or x = -7 included in the solution?

Because our inequality uses > (greater than), not ≥. At x = 1: $|4(1)+12| = 16$ , which equals 16 but doesn't satisfy > 16.

How can I check if my solution regions are correct?

Pick a test point from each region (like x = 2 and x = -8)
Substitute into the original inequality
Both should make $|4x+12| > 16$ true
Pick a point between the regions (like x = 0) - it should make the inequality false

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Solve |4x+12| > 16: Absolute Value Inequality Challenge

Absolute Value Inequalities with Union Solutions

❤️ 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 get two separate inequalities from one absolute value inequality?

How do I know when to use 'or' versus 'and' in my final answer?

What does the solution x > 1 or x < -7 look like on a number line?

Why isn't x = 1 or x = -7 included in the solution?

How can I check if my solution regions are correct?

🌟 Unlock Your Math Potential

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