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

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

Because absolute value means distance from zero, which is always positive! When $ |A| = |B| $, either $ A = B $ (same direction) or $ A = -B $ (opposite directions).

Q: How do I know which case to solve first?

It doesn't matter which case you solve first! Always solve both cases completely, then check which solutions satisfy any additional constraints like \( x .

Q: What if both solutions satisfy the constraint?

Then you would have two valid solutions! But in this problem, only $ x = -16 $ satisfies $ x because \( -8 > -10 $.

Q: Can I solve this by squaring both sides instead?

Yes, but be careful! Squaring gives $ (x+4)^2 = (2x+20)^2 $, which leads to the same two solutions. However, the case method is clearer and less prone to algebraic errors.

Q: Why is the constraint x < -10 important?

The constraint limits which solutions are valid. Without it, both $ x = -16 $ and $ x = -8 $ would be correct. Real-world problems often have such restrictions!

Step-by-step video solution

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Step-by-step written solution

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1

Understand the problem

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

2

Step-by-step solution

To solve the given problem, we'll work through these steps:

Step 1: Solve the equation $|x+4| = |2x+20|$ .
Step 2: Consider the two cases derived from the absolute value equation.
Step 3: Apply the condition $x < -10$ to determine the suitable solution.

Step 1: We start by considering the absolute value equation $|x+4| = |2x+20|$ . This gives us two cases to explore:

Case 1: $x+4 = 2x+20$
Case 2: $x+4 = -(2x+20)$

Step 2: Solve each case individually:

Case 1:
Solving $x+4 = 2x+20$ ,
We first subtract $x$ from both sides to obtain:
$4 = x + 20$ .
Subtracting 20 from both sides, we get:
$x = 4 - 20 \Rightarrow x = -16$ .

Case 2:
Solving $x+4 = -(2x+20)$ ,
This simplifies to $x+4 = -2x - 20$ .
Adding $2x$ to both sides, we have:
$3x + 4 = -20$ .
Subtracting 4 from both sides gives:
$3x = -24$ .
Dividing by 3, we find:
$x = -8$ .

Step 3: Consider the inequality $x < -10$ :

Substitute $x = -16$ into the inequality: $-16 < -10$ is true.
Substitute $x = -8$ into the inequality: $-8 < -10$ is false.

Therefore, the solution that satisfies both the equation and the inequality is $x = -16$ .

3

Final Answer

$x=-16$

Key Points to Remember

Essential concepts to master this topic

Two Cases Rule: Set expressions equal and opposite for absolute value equations
Case Solving: $x+4 = 2x+20$ gives $x = -16$ ; $x+4 = -(2x+20)$ gives $x = -8$
Constraint Check: Test both solutions against $x < -10$ : only $x = -16$ satisfies ✓

Common Mistakes

Avoid these frequent errors

Forgetting to check solutions against the given constraint
Don't solve $|x+4| = |2x+20|$ and pick any solution = wrong answer! This ignores the $x < -10$ condition. Both $x = -16$ and $x = -8$ satisfy the equation, but only $x = -16$ satisfies $x < -10$ . Always test each solution against all given constraints.

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?

+

Because absolute value means distance from zero, which is always positive! When $|A| = |B|$ , either $A = B$ (same direction) or $A = -B$ (opposite directions).

How do I know which case to solve first?

+

It doesn't matter which case you solve first! Always solve both cases completely, then check which solutions satisfy any additional constraints like $x < -10$ .

What if both solutions satisfy the constraint?

+

Then you would have two valid solutions! But in this problem, only $x = -16$ satisfies $x < -10$ because $-8 > -10$ .

Can I solve this by squaring both sides instead?

+

Yes, but be careful! Squaring gives $(x+4)^2 = (2x+20)^2$ , which leads to the same two solutions. However, the case method is clearer and less prone to algebraic errors.

Why is the constraint x < -10 important?

+

The constraint limits which solutions are valid. Without it, both $x = -16$ and $x = -8$ would be correct. Real-world problems often have such restrictions!

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

Absolute Value Equations with Inequality Constraints

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