Solve the Double Absolute Value Equation: |-x| = 10

Absolute Value Equations with Negated Variables

∣−x∣=10 \left|-x\right|=10

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1

Understand the problem

∣−x∣=10 \left|-x\right|=10

2

Step-by-step solution

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

  • Step 1: Use the definition of absolute value to create equations.
  • Step 2: Solve each equation to find possible values of x x .
  • Step 3: Check the solutions against the original problem.

Now, let's work through each step:
Step 1: The equation is given as ∣−x∣=10 \left| -x \right| = 10 . According to the definition of absolute value:

  • Case 1: −x=10-x = 10
  • Case 2: −x=−10-x = -10

Step 2: Solve each case:
In Case 1, we have:

  • −x=10-x = 10
    This implies x=−10 x = -10 .

In Case 2, we have:

  • −x=−10-x = -10
    This implies x=10 x = 10 .

Step 3: Therefore, the possible solutions are x=−10 x = -10 and x=10 x = 10 , which satisfy the equation independently.

The solution to the problem is:

x=−10 x = -10 , x=10 x = 10

3

Final Answer

x=−10 x=-10 , x=10 x=10

Key Points to Remember

Essential concepts to master this topic
  • Definition: Absolute value equals distance from zero, always positive
  • Cases: ∣−x∣=10 |-x| = 10 gives -x = 10 and -x = -10
  • Check: Both x = -10 and x = 10 make ∣−x∣=10 |-x| = 10 true ✓

Common Mistakes

Avoid these frequent errors
  • Forgetting the second case when solving absolute value equations
    Don't only solve -x = 10 to get x = -10! This misses half the solution because absolute value creates two cases. Always set up both -x = 10 AND -x = -10 to find all solutions.

Practice Quiz

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\( \left|-x\right|=10 \)

FAQ

Everything you need to know about this question

Why does |-x| = 10 have two solutions?

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Because absolute value measures distance from zero! Both 10 and -10 are exactly 10 units away from zero on the number line, so when we have -x inside, both x = 10 and x = -10 work.

Is |-x| the same as |x|?

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Yes! Since |-x| = |x| for any value of x, they're identical. The negative sign inside doesn't change the absolute value result, but it affects how we solve the equation.

How do I set up the two cases?

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For ∣expression∣=number |expression| = number , create two equations:

  • expression=number expression = number
  • expression=−number expression = -number

Then solve both separately!

Do I always get two solutions for absolute value equations?

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Usually yes, but not always! Sometimes both cases give the same answer, or one case might not work. That's why it's important to check both solutions in the original equation.

What if I get confused about the negative signs?

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Take it step by step! For ∣−x∣=10 |-x| = 10 , just remember that the expression inside the absolute value bars (-x) can equal either +10 or -10. Solve each case carefully.

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