Solve the Absolute Value Equation: Find x in |x - 3| = 4

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

Absolute value measures distance, which is always positive. Since both 4 and -4 are distance 4 from zero, both $ x = 7 $ and $ x = -1 $ work!

Q: How do I know which equation to set up first?

It doesn't matter! You need both equations: one where the inside equals the positive value, and one where it equals the negative value. Always do both!

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

Yes, but it's trickier! Squaring gives $ (x-3)^2 = 16 $, which you'd solve as a quadratic. The two-equation method is usually faster and clearer.

Q: How do I check if both answers are correct?

Substitute each solution back into the original equation:For x = -1: $ |-1-3| = |-4| = 4 $ ✓For x = 7: $ |7-3| = |4| = 4 $ ✓

Absolute Value Equations with Two Solutions

$\left|x-3\right|=4$

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Understand the problem

$\left|x-3\right|=4$

Step-by-step solution

To solve the equation $|x - 3| = 4$ , follow these steps:

Step 1: Understand that $|x - 3| = 4$ means $x - 3$ can be either 4 or -4.
Step 2: Set up two separate equations:

Equation 1: $x - 3 = 4$
Equation 2: $x - 3 = -4$

Step 3: Solve each equation for $x$ .

Let's solve Equation 1:
$x - 3 = 4$
Add 3 to both sides:
$x = 4 + 3$
$x = 7$

Now, solve Equation 2:
$x - 3 = -4$
Add 3 to both sides:
$x = -4 + 3$
$x = -1$

Therefore, the solutions to the equation $|x - 3| = 4$ are $x = 7$ and $x = -1$ .

Checking the given choices, the correct answer is:
$x = -1$ , $x = 7$ , which matches choice 1.

Final Answer

$x=-1$ , $x=7$

Key Points to Remember

Essential concepts to master this topic

Definition: |x - 3| = 4 means distance from 3 equals 4
Technique: Set up two equations: x - 3 = 4 and x - 3 = -4
Check: Substitute both solutions: |-1 - 3| = 4 ✓ and |7 - 3| = 4 ✓

Common Mistakes

Avoid these frequent errors

Only solving one equation and missing the second solution
Don't solve just x - 3 = 4 and stop = only getting x = 7! This misses half the answer because absolute value creates two possibilities. Always set up both positive and negative cases to find both solutions.

Practice Quiz

Test your knowledge with interactive questions

$\left|x\right|=5$

FAQ

Everything you need to know about this question

Why does an absolute value equation have two solutions?

Absolute value measures distance, which is always positive. Since both 4 and -4 are distance 4 from zero, both $x = 7$ and $x = -1$ work!

How do I know which equation to set up first?

It doesn't matter! You need both equations: one where the inside equals the positive value, and one where it equals the negative value. Always do both!

What if I get the same answer from both equations?

That's impossible with this type of problem! If you get the same answer twice, check your arithmetic. Different equations should give different solutions.

Can I solve this by squaring both sides instead?

Yes, but it's trickier! Squaring gives $(x-3)^2 = 16$ , which you'd solve as a quadratic. The two-equation method is usually faster and clearer.

How do I check if both answers are correct?

Substitute each solution back into the original equation:

For x = -1: $|-1-3| = |-4| = 4$ ✓
For x = 7: $|7-3| = |4| = 4$ ✓

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