Solving the Absolute Value Equation: Discover x When |x - 1| = 6

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

Because absolute value measures distance from zero! Both $ x = -5 $ and $ x = 7 $ are exactly 6 units away from 1 on the number line.

Q: How do I know which equation to write first?

It doesn't matter! For $ |x - 1| = 6 $, you can write $ x - 1 = 6 $ or $ x - 1 = -6 $ first. Just make sure you write both equations.

Q: What if I get the same answer from both equations?

That means your absolute value equation has only one solution! This happens when the expression inside equals zero. Always check both equations even if they look similar.

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

Yes, but it's more complicated! Squaring gives $ (x-1)^2 = 36 $, then you need to expand and solve a quadratic. The two-equation method is simpler for basic absolute value problems.

Q: How do I check if my solutions are correct?

Substitute each solution back into the original equation:For $ x = -5 $: $ |-5 - 1| = |-6| = 6 $ ✓For $ x = 7 $: $ |7 - 1| = |6| = 6 $ ✓

Absolute Value Equations with Two Solutions

$\left|x-1\right|=6$

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Understand the problem

$\left|x-1\right|=6$

Step-by-step solution

To solve the equation $|x-1| = 6$ , we will use the definition of absolute value to create two separate linear equations:

Equation 1: $x - 1 = 6$
Equation 2: $x - 1 = -6$

Let's solve each equation separately:

For the first equation $x - 1 = 6$ :

Add 1 to both sides of the equation:
$x - 1 + 1 = 6 + 1$
This simplifies to $x = 7$ .

For the second equation $x - 1 = -6$ :

Add 1 to both sides of the equation:
$x - 1 + 1 = -6 + 1$
This simplifies to $x = -5$ .

Thus, the solutions to the equation $|x-1| = 6$ are $x = -5$ and $x = 7$ .

Therefore, the correct solutions are $x = -5$ and $x = 7$ .

Final Answer

$x=-5$ , $x=7$

Key Points to Remember

Essential concepts to master this topic

Definition: Absolute value equation creates two separate linear equations
Method: Set x - 1 = 6 and x - 1 = -6
Verification: Check both solutions: |-5 - 1| = 6 and |7 - 1| = 6 ✓

Common Mistakes

Avoid these frequent errors

Solving only one equation or forgetting the negative case
Don't solve just x - 1 = 6 and ignore x - 1 = -6 = only one solution instead of two! Absolute value means distance, so both positive and negative values work. Always create and solve both equations from the absolute value definition.

Practice Quiz

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$\left|x\right|=5$

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 $x = -5$ and $x = 7$ are exactly 6 units away from 1 on the number line.

How do I know which equation to write first?

It doesn't matter! For $|x - 1| = 6$ , you can write $x - 1 = 6$ or $x - 1 = -6$ first. Just make sure you write both equations.

What if I get the same answer from both equations?

That means your absolute value equation has only one solution! This happens when the expression inside equals zero. Always check both equations even if they look similar.

Can I solve this by squaring both sides instead?

Yes, but it's more complicated! Squaring gives $(x-1)^2 = 36$ , then you need to expand and solve a quadratic. The two-equation method is simpler for basic absolute value problems.

How do I check if my solutions are correct?

Substitute each solution back into the original equation:

For $x = -5$ : $|-5 - 1| = |-6| = 6$ ✓
For $x = 7$ : $|7 - 1| = |6| = 6$ ✓

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