Solving the Absolute Value Puzzle: |5 - x| = 7

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

Because absolute value measures distance from zero, both positive and negative numbers can have the same absolute value. For example, both 7 and -7 are 7 units from zero, so $ |7| = |-7| = 7 $.

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

It doesn't matter! You can solve either Case 1 or Case 2 first. Just make sure you solve both cases to find all possible solutions.

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

Substitute each solution back into the original equation. For $ x = -2 $: $ |5 - (-2)| = |7| = 7 $ ✓For $ x = 12 $: $ |5 - 12| = |-7| = 7 $ ✓

Q: Can absolute value equations have no solutions?

Yes! If you have something like $ |x| = -5 $, there are no solutions because absolute values are never negative. Always check that the number on the right side is positive or zero.

Absolute Value Equations with Two Solutions

$|5 - x| = 7$

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$|5 - x| = 7$

Step-by-step solution

To solve this problem, we'll use the absolute value property, which splits the equation into two separate cases:

Case 1: Solve $5 - x = 7$

Subtract 5 from both sides:

$5 - x - 5 = 7 - 5$

Which simplifies to:

$-x = 2$

Multiplying both sides by -1 gives:

$x = -2$

Case 2: Solve $5 - x = -7$

Subtract 5 from both sides:

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

Which simplifies to:

$-x = -12$

Multiplying both sides by -1 gives:

$x = 12$

Thus, the solutions are $x = -2$ and $x = 12$ . Now let's verify against the available choices. We can see that:

x = 0
x = -2
x = 12
Answers b + c (i.e., x = -2 and x = 12)

This confirms that the correct choice is "Answers b + c".

Therefore, the correct answer is Answers b + c.

Final Answer

Answers b + c

Key Points to Remember

Essential concepts to master this topic

Definition: |expression| = positive number creates two separate equations
Method: Set 5 - x = 7 and 5 - x = -7
Verification: Check both solutions: |5 - (-2)| = |7| = 7 ✓ and |5 - 12| = |-7| = 7 ✓

Common Mistakes

Avoid these frequent errors

Solving only one case of the absolute value equation
Don't just solve 5 - x = 7 and stop there = missing half the solutions! Absolute value equations always have two cases because the expression inside could be positive or negative. Always set up both cases: expression = positive value AND expression = negative value.

Practice Quiz

Test your knowledge with interactive questions

$\left|x\right|=3$

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 positive and negative numbers can have the same absolute value. For example, both 7 and -7 are 7 units from zero, so $|7| = |-7| = 7$ .

How do I know which case to solve first?

It doesn't matter! You can solve either Case 1 or Case 2 first. Just make sure you solve both cases to find all possible solutions.

What if I get the same answer from both cases?

That can happen! Sometimes both cases give you the same solution, which means there's only one unique answer. Always solve both cases anyway to be thorough.

How do I check if my solutions are correct?

Substitute each solution back into the original equation. For $x = -2$ : $|5 - (-2)| = |7| = 7$ ✓
For $x = 12$ : $|5 - 12| = |-7| = 7$ ✓

Can absolute value equations have no solutions?

Yes! If you have something like $|x| = -5$ , there are no solutions because absolute values are never negative. Always check that the number on the right side is positive or zero.

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