Solve the Absolute Value Equation: |x + 2| = 4

Q: Why does an absolute value equation have two solutions?

Because absolute value measures distance from zero! Both -4 and 4 are exactly 4 units from zero, so $ |\text{something}|=4 $ means that 'something' could be 4 OR -4.

Q: What if I get only one solution?

Double-check your work! Most absolute value equations have two solutions. Make sure you solved both cases: the positive case AND the negative case.

Q: How can I verify both answers are correct?

Substitute each solution back into the original equation. For x=-6: $ |-6+2|=|-4|=4 $ ✓. For x=2: $ |2+2|=|4|=4 $ ✓.

Q: Do absolute value equations always have exactly two solutions?

Usually, but not always! They can have two solutions (most common), one solution (when the expression equals zero), or no solutions (when equal to a negative number).

Absolute Value Equations with Two Solutions

$\left|x+2\right|=4$

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Understand the problem

$\left|x+2\right|=4$

Step-by-step solution

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

Step 1: Set up two linear equations from the absolute value definition.
Step 2: Solve each equation for $x$ .

Now, let's work through each step:

Step 1: Consider the two cases from $\left|x + 2\right| = 4$ .

First case: $x + 2 = 4$ .
Second case: $x + 2 = -4$ .

Step 2: Solve the two equations:

For the first case:

x + 2 = 4

Subtract 2 from both sides:

x = 4 - 2

x = 2

For the second case:

x + 2 = -4

Subtract 2 from both sides:

x = -4 - 2

x = -6

Therefore, the solutions to the equation $\left|x + 2\right| = 4$ are:

$x = -6$ and $x = 2$ .

The correct option is:

$x = -6, x = 2$

Thus, the solution to the problem is:

$x = -6, x = 2$

Final Answer

$x=-6$ , $x=2$

Key Points to Remember

Essential concepts to master this topic

Definition: Absolute value creates two cases: positive and negative expressions
Technique: From $|x+2|=4$ get x+2=4 and x+2=-4
Check: Substitute both solutions: $|-6+2|=4$ and $|2+2|=4$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting the negative case
Don't solve only x+2=4 and get just x=2! This misses half the solution because absolute value means distance from zero. Always solve both x+2=4 AND x+2=-4 to find both solutions.

Practice Quiz

Test your knowledge with interactive questions

$\left|-x\right|=10$

FAQ

Everything you need to know about this question

Why does an absolute value equation have two solutions?

Because absolute value measures distance from zero! Both -4 and 4 are exactly 4 units from zero, so $|\text{something}|=4$ means that 'something' could be 4 OR -4.

How do I set up the two cases?

Take what's inside the absolute value bars and set it equal to both the positive and negative values. So $|x+2|=4$ becomes: x+2=4 and x+2=-4.

What if I get only one solution?

Double-check your work! Most absolute value equations have two solutions. Make sure you solved both cases: the positive case AND the negative case.

How can I verify both answers are correct?

Substitute each solution back into the original equation. For x=-6: $|-6+2|=|-4|=4$ ✓. For x=2: $|2+2|=|4|=4$ ✓.

Do absolute value equations always have exactly two solutions?

Usually, but not always! They can have two solutions (most common), one solution (when the expression equals zero), or no solutions (when equal to a negative number).

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