Solve the Absolute Value Inequality: |x + 3| ≤ 5

Q: Why does $ |x + 3| \leq 5 $ become a compound inequality?

Because absolute value measures distance from zero! When $ |A| \leq 5 $, it means A is within 5 units of zero, so A can be anywhere from -5 to 5.

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

Think of it as "sandwiching" the expression! $ |A| \leq B $ means A is trapped between -B and B: $ -B \leq A \leq B $.

Q: What if I get $ |x + 3| \geq 5 $ instead?

Then you'd have two separate inequalities: $ x + 3 \leq -5 $ OR $ x + 3 \geq 5 $. The solution would be two rays, not an interval!

Q: How can I check my answer $ -8 \leq x \leq 2 $ ?

Test the boundary values and a middle value! Try $ x = -8 $: $ |-8 + 3| = 5 \leq 5 $ ✓. Try $ x = 0 $: $ |0 + 3| = 3 \leq 5 $ ✓.

Q: What does the solution interval $ -8 \leq x \leq 2 $ look like on a number line?

It's a closed interval from -8 to 2, including both endpoints. Draw a solid line segment with filled circles at -8 and 2!

Absolute Value Inequalities with Linear Expressions

Find the absolute value inequality representation for:

$|x + 3| \leq 5$

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Find the absolute value inequality representation for:

$|x + 3| \leq 5$

Step-by-step solution

To solve the inequality $|x + 3| \leq 5$ , we first consider the definition of absolute value inequality $|A| \leq B$ , which is equivalent to $-B \leq A \leq B$ .

Applying this definition, we have:

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

Next, we isolate $x$ by subtracting $3$ from all parts of the inequality:

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

This simplifies to:

$-8 \leq x \leq 2$

Final Answer

$-8 \leq x \leq 2$

Key Points to Remember

Essential concepts to master this topic

Rule: $|A| \leq B$ becomes $-B \leq A \leq B$
Technique: Apply definition: $-5 \leq x + 3 \leq 5$ , then subtract 3
Check: Test boundary values: $|(-8) + 3| = |-5| = 5 \leq 5$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting to apply the definition to both sides
Don't solve $|x + 3| \leq 5$ as just $x + 3 \leq 5$ = $x \leq 2$ ! This ignores the negative case and gives only half the solution. Always write the compound inequality $-5 \leq x + 3 \leq 5$ first.

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 $|x + 3| \leq 5$ become a compound inequality?

Because absolute value measures distance from zero! When $|A| \leq 5$ , it means A is within 5 units of zero, so A can be anywhere from -5 to 5.

How do I remember which direction the inequality signs go?

Think of it as "sandwiching" the expression! $|A| \leq B$ means A is trapped between -B and B: $-B \leq A \leq B$ .

What if I get $|x + 3| \geq 5$ instead?

Then you'd have two separate inequalities: $x + 3 \leq -5$ OR $x + 3 \geq 5$ . The solution would be two rays, not an interval!

How can I check my answer $-8 \leq x \leq 2$ ?

Test the boundary values and a middle value! Try $x = -8$ : $|-8 + 3| = 5 \leq 5$ ✓. Try $x = 0$ : $|0 + 3| = 3 \leq 5$ ✓.

What does the solution interval $-8 \leq x \leq 2$ look like on a number line?

It's a closed interval from -8 to 2, including both endpoints. Draw a solid line segment with filled circles at -8 and 2!

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Solve the Absolute Value Inequality: |x + 3| ≤ 5

Absolute Value Inequalities with Linear Expressions

❤️ 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 $|x + 3| \leq 5$ become a compound inequality?

How do I remember which direction the inequality signs go?

What if I get $|x + 3| \geq 5$ instead?

How can I check my answer $-8 \leq x \leq 2$ ?

What does the solution interval $-8 \leq x \leq 2$ look like on a number line?

🌟 Unlock Your Math Potential

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Solve the Absolute Value Inequality: |x + 3| ≤ 5

Absolute Value Inequalities with Linear Expressions

❤️ 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 ∣x+3∣≤5 |x + 3| \leq 5 ∣x+3∣≤5 become a compound inequality?

How do I remember which direction the inequality signs go?

What if I get ∣x+3∣≥5 |x + 3| \geq 5 ∣x+3∣≥5 instead?

How can I check my answer −8≤x≤2 -8 \leq x \leq 2 −8≤x≤2?

What does the solution interval −8≤x≤2 -8 \leq x \leq 2 −8≤x≤2 look like on a number line?

🌟 Unlock Your Math Potential

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Why does $|x + 3| \leq 5$ become a compound inequality?

What if I get $|x + 3| \geq 5$ instead?

How can I check my answer $-8 \leq x \leq 2$ ?

What does the solution interval $-8 \leq x \leq 2$ look like on a number line?