Solve the Absolute Value Equation System: |x+4| = |2x+20| with x > -10

Q: What does x > -10 mean for my solutions?

This is a domain restriction - it limits which x-values are allowed. Even if you find a solution algebraically, you must reject it if it doesn't satisfy x > -10.

Q: How do I know which case to use when solving?

You don't choose - you must solve both cases! Then check which solutions (if any) satisfy the domain constraint x > -10.

Q: Can I have zero solutions or multiple solutions?

Absolutely! Absolute value equations can have 0, 1, or 2 solutions depending on the constraint. In this problem, only one of our two algebraic solutions satisfies x > -10.

Q: How do I verify my final answer?

Substitute x = -8 into the original equation: $|-8+4| = |2(-8)+20|$ becomes $|-4| = |4|$, which gives $4 = 4$ ✓

Absolute Value Equations with Domain Restrictions

$\begin{cases} |x+4|=|2x+20| \\ x> -10 \end{cases}$

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Understand the problem

$\begin{cases} |x+4|=|2x+20| \\ x> -10 \end{cases}$

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Consider the equation $|x+4| = |2x+20|$ .
Step 2: Break it down into two separate cases.

We'll address each case separately below:

Case 1: $x + 4 = 2x + 20$

Simplifying gives:

$x + 4 = 2x + 20$
$4 = 2x - x + 20$
$4 = x + 20$
Subtracting 20 from both sides, we get:
$x = 4 - 20$
$x = -16$ .

Since $x > -10$ is required, $x = -16$ is not valid.

Case 2: $x + 4 = -(2x + 20)$

Simplifying gives:

$x + 4 = -2x - 20$
Add $2x$ to both sides:
$x + 2x + 4 = -20$
$3x + 4 = -20$
Subtract 4 from both sides:
$3x = -24$
Divide by 3:
$x = -8$ .

Check: Since $-8 > -10$ , this solution is valid.

Thus, the solution to the problem is $x = -8$ .

Final Answer

$x=-8$

Key Points to Remember

Essential concepts to master this topic

Rule: Solve |A| = |B| by considering both A = B and A = -B
Technique: For |x+4| = |2x+20|, solve x+4 = 2x+20 and x+4 = -(2x+20)
Check: Verify solution satisfies constraint: -8 > -10 ✓

Common Mistakes

Avoid these frequent errors

Forgetting to check the domain constraint
Don't solve |x+4| = |2x+20| and accept x = -16 without checking x > -10! This violates the given constraint and gives an invalid solution. Always verify each solution satisfies all given conditions, especially domain restrictions.

Practice Quiz

Test your knowledge with interactive questions

$\left|x\right|=3$

FAQ

Everything you need to know about this question

Why do I need to consider two cases for absolute value equations?

The absolute value |A| equals either A (when A ≥ 0) or -A (when A < 0). So |A| = |B| means either A = B or A = -B. You must check both possibilities!

What does x > -10 mean for my solutions?

This is a domain restriction - it limits which x-values are allowed. Even if you find a solution algebraically, you must reject it if it doesn't satisfy x > -10.

How do I know which case to use when solving?

You don't choose - you must solve both cases! Then check which solutions (if any) satisfy the domain constraint x > -10.

Can I have zero solutions or multiple solutions?

Absolutely! Absolute value equations can have 0, 1, or 2 solutions depending on the constraint. In this problem, only one of our two algebraic solutions satisfies x > -10.

How do I verify my final answer?

Substitute x = -8 into the original equation: $|-8+4| = |2(-8)+20|$ becomes $|-4| = |4|$ , which gives $4 = 4$ ✓

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