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

Q: Why does the absolute value inequality split into two cases?

The absolute value |x-5| represents the distance from x to 5. When this distance is greater than 11, x can be either more than 11 units to the right of 5 (x > 16) or more than 11 units to the left of 5 (x

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

Think of it this way: if |A| > B, then A is either bigger than B (A > B) or smaller than the opposite of B (A

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

With greater than (>), the solution is OUTSIDE the boundary points (use OR). With less than (, the solution would be INSIDE the boundary points (use AND). Remember: > means 'far from center',

Q: How can I check if my answer makes sense?

Pick test values from your solution regions! For example, try x = 20: |20-5| = 15, and 15 > 11 ✓. Try x = -10: |-10-5| = 15, and 15 > 11 ✓. Both work!

Q: What if I get confused about which boundary values to use?

Always solve the two cases step by step: x-5 > 11 gives x > 16, and x-5 gives x

Absolute Value Inequalities with Greater Than

Given:

$\left|x-5\right|>11$

Which of the following statements is necessarily true?

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Understand the problem

Given:

$\left|x-5\right|>11$

Which of the following statements is necessarily true?

Step-by-step solution

To solve the inequality $\left|x-5\right| > 11$ , we first apply the property of absolute values:

If $\left|A\right| > B$ , then $A > B$ or $A < -B$ .

Therefore, for $\left|x-5\right| > 11$ , we have two cases to consider:

Case 1: $x-5 > 11$
Case 2: $x-5 < -11$

Let's solve each case separately:

Case 1: $x-5 > 11$

Add 5 to both sides to isolate $x$ :
$x > 11 + 5$

This simplifies to:

$x > 16$

Case 2: $x-5 < -11$

Add 5 to both sides to isolate $x$ :
$x < -11 + 5$

This simplifies to:

$x < -6$

Thus, the solution to the inequality is:

$x > 16$ or $x < -6$

Comparing this result with the given answer choices, the correct one is:

$x>16$ o $x<-6$

Therefore, the solution to the problem is $x > 16$ or $x < -6$ .

Final Answer

$x>16$ or $x<-6$

Key Points to Remember

Essential concepts to master this topic

Rule: When |A| > B, then A > B or A < -B
Technique: Solve x-5 > 11 gives x > 16; x-5 < -11 gives x < -6
Check: Test x = 20: |20-5| = 15 > 11 ✓ and x = -10: |-10-5| = 15 > 11 ✓

Common Mistakes

Avoid these frequent errors

Writing the solution as a compound inequality with AND
Don't write -6 < x < 16 or think the solution is between the boundary values! This gives the opposite solution set. Always remember that |x-5| > 11 means the distance is GREATER than 11, so x must be OUTSIDE the interval [-6, 16].

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 does the absolute value inequality split into two cases?

The absolute value |x-5| represents the distance from x to 5. When this distance is greater than 11, x can be either more than 11 units to the right of 5 (x > 16) or more than 11 units to the left of 5 (x < -6).

How do I remember which direction the inequality signs go?

Think of it this way: if |A| > B, then A is either bigger than B (A > B) or smaller than the opposite of B (A < -B). The absolute value 'splits' the inequality in both directions!

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

With greater than (>), the solution is OUTSIDE the boundary points (use OR). With less than (<), the solution would be INSIDE the boundary points (use AND). Remember: > means 'far from center', < means 'close to center'.

How can I check if my answer makes sense?

Pick test values from your solution regions! For example, try x = 20: |20-5| = 15, and 15 > 11 ✓. Try x = -10: |-10-5| = 15, and 15 > 11 ✓. Both work!

What if I get confused about which boundary values to use?

Always solve the two cases step by step: x-5 > 11 gives x > 16, and x-5 < -11 gives x < -6. Don't try to memorize - just follow the algebra carefully each time!

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

Absolute Value Inequalities with Greater Than

❤️ 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 does the absolute value inequality split into two cases?

How do I remember which direction the inequality signs go?

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

How can I check if my answer makes sense?

What if I get confused about which boundary values to use?

🌟 Unlock Your Math Potential

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