Solve |x-3| ≤ 5: Absolute Value Inequality Step-by-Step

Q: Why does the absolute value inequality become a compound inequality with AND?

Because $ |x-3| \leq 5 $ means the distance from x to 3 is at most 5 units. This creates a bounded interval where x must satisfy both conditions simultaneously!

Q: How is this different from $ |x-3| \geq 5 $ ?

Great question! $ |x-3| \geq 5 $ would give you two separate regions with OR: x ≤ -2 OR x ≥ 8. The ≤ keeps values between the boundaries, while ≥ takes values outside them.

Q: What if I get confused about which direction the inequality goes?

Think about it step by step: $ |x-3| \leq 5 $ means $ -5 \leq x-3 \leq 5 $. Then add 3 to all three parts: $ -5+3 \leq x \leq 5+3 $, so $ -2 \leq x \leq 8 $.

Q: What happens if the right side was negative, like $ |x-3| \leq -2 $ ?

That would have no solution! Absolute values are always non-negative (≥ 0), so they can never be less than or equal to a negative number. Watch out for this trap!

Absolute Value Inequalities with Compound Solutions

Given:

$|x-3| \leq 5$

Which of the following statements is necessarily true?

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Given:

$|x-3| \leq 5$

Which of the following statements is necessarily true?

Step-by-step solution

To solve the inequality $|x-3| \leq 5$ , we need to consider the definition of absolute value inequalities. The inequality $|a| \leq b$ translates to $-b \leq a \leq b$ .

Applying this to our expression $|x-3| \leq 5$ , we have:

$-5 \leq x-3 \leq 5$ .

We add 3 to all parts of the inequality to isolate $x$ :

$-5 + 3 \leq x - 3 + 3 \leq 5 + 3$

This simplifies to $-2 \leq x \leq 8$ .

Final Answer

$-2 \leq x \leq 8$

Key Points to Remember

Essential concepts to master this topic

Rule: $|a| \leq b$ becomes $-b \leq a \leq b$
Technique: Rewrite $|x-3| \leq 5$ as $-5 \leq x-3 \leq 5$
Check: Test boundary values: $|-2-3| = 5 \leq 5$ and $|8-3| = 5 \leq 5$ ✓

Common Mistakes

Avoid these frequent errors

Converting to two separate inequalities with OR
Don't split $|x-3| \leq 5$ into x-3 ≤ 5 OR x-3 ≥ -5 = wrong solution set! This gives you all real numbers instead of the bounded interval. Always use AND with compound inequalities: $-5 \leq x-3 \leq 5$ .

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 become a compound inequality with AND?

Because $|x-3| \leq 5$ means the distance from x to 3 is at most 5 units. This creates a bounded interval where x must satisfy both conditions simultaneously!

How is this different from $|x-3| \geq 5$ ?

Great question! $|x-3| \geq 5$ would give you two separate regions with OR: x ≤ -2 OR x ≥ 8. The ≤ keeps values between the boundaries, while ≥ takes values outside them.

What if I get confused about which direction the inequality goes?

Think about it step by step: $|x-3| \leq 5$ means $-5 \leq x-3 \leq 5$ . Then add 3 to all three parts: $-5+3 \leq x \leq 5+3$ , so $-2 \leq x \leq 8$ .

How can I visualize this on a number line?

Draw a number line and mark points -2 and 8. Since we have ≤, use closed circles (filled dots) at both endpoints and shade the entire region between them. This shows all valid x values!

What happens if the right side was negative, like $|x-3| \leq -2$ ?

That would have no solution! Absolute values are always non-negative (≥ 0), so they can never be less than or equal to a negative number. Watch out for this trap!

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Solve |x-3| ≤ 5: Absolute Value Inequality Step-by-Step

Absolute Value Inequalities with Compound Solutions

❤️ Continue Your Math Journey!

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 become a compound inequality with AND?

How is this different from $|x-3| \geq 5$ ?

What if I get confused about which direction the inequality goes?

How can I visualize this on a number line?

What happens if the right side was negative, like $|x-3| \leq -2$ ?

🌟 Unlock Your Math Potential

More Questions

Inequalities with Absolute Values

Solve |2x + 1| > 7: Absolute Value Inequality Challenge

Solve |x+5| < 2: Absolute Value Inequality Analysis

Solve the Nested Absolute Value Inequality: |a|-||5-4|-1| > 0

Solve the Inequality: |−4 + 8| − 2|−|a| > 0

Solve |x-3| ≤ 5: Absolute Value Inequality Step-by-Step

Absolute Value Inequalities with Compound Solutions

❤️ Continue Your Math Journey!

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 become a compound inequality with AND?

How is this different from ∣x−3∣≥5 |x-3| \geq 5 ∣x−3∣≥5?

What if I get confused about which direction the inequality goes?

How can I visualize this on a number line?

What happens if the right side was negative, like ∣x−3∣≤−2 |x-3| \leq -2 ∣x−3∣≤−2?

🌟 Unlock Your Math Potential

More Questions

Inequalities with Absolute Values

Solve |2x + 1| > 7: Absolute Value Inequality Challenge

Solve |x+5| < 2: Absolute Value Inequality Analysis

Solve the Nested Absolute Value Inequality: |a|-||5-4|-1| > 0

Solve the Inequality: |−4 + 8| − 2|−|a| > 0

How is this different from $|x-3| \geq 5$ ?

What happens if the right side was negative, like $|x-3| \leq -2$ ?