How To Solve Absolute Value Equations, Basic Introduction, Algebra - Free Printable

Let's solve each of the absolute value equations step by step. Absolute value equations involve expressions inside absolute value bars, and they are solved by considering both the positive and negative possibilities.

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1. $ |x| = 7 $

The absolute value of a number is its distance from zero on the number line, so:

$$
|x| = 7 \Rightarrow x = 7 \quad \text{or} \quad x = -7
$$

✔ Solutions: $ x = 7 $ or $ x = -7 $

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2. $ |x + 2| = 9 $

We set up two equations based on the definition of absolute value:

$$
x + 2 = 9 \quad \text{or} \quad x + 2 = -9
$$

Solve each:

- $ x + 2 = 9 \Rightarrow x = 7 $
- $ x + 2 = -9 \Rightarrow x = -11 $

✔ Solutions: $ x = 7 $ or $ x = -11 $

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3. $ |2x - 5| = 1 $

Set up two cases:

$$
2x - 5 = 1 \quad \text{or} \quad 2x - 5 = -1
$$

Solve each:

- $ 2x - 5 = 1 \Rightarrow 2x = 6 \Rightarrow x = 3 $
- $ 2x - 5 = -1 \Rightarrow 2x = 4 \Rightarrow x = 2 $

✔ Solutions: $ x = 3 $ or $ x = 2 $

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4. $ 3|x - 1| - 2 = 7 $

First, isolate the absolute value expression.

Add 2 to both sides:

$$
3|x - 1| = 9
$$

Divide both sides by 3:

$$
|x - 1| = 3
$$

Now solve:

$$
x - 1 = 3 \quad \text{or} \quad x - 1 = -3
$$

- $ x - 1 = 3 \Rightarrow x = 4 $
- $ x - 1 = -3 \Rightarrow x = -2 $

✔ Solutions: $ x = 4 $ or $ x = -2 $

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✔ Final Answers:

1. $ |x| = 7 $ → $ x = \boxed{-7,\ 7} $
2. $ |x + 2| = 9 $ → $ x = \boxed{-11,\ 7} $
3. $ |2x - 5| = 1 $ → $ x = \boxed{2,\ 3} $
4. $ 3|x - 1| - 2 = 7 $ → $ x = \boxed{-2,\ 4} $

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🔍 Key Concept:

For any equation $ |A| = B $ (where $ B \geq 0 $), the solutions are:
$$
A = B \quad \text{or} \quad A = -B
$$
Always check that the right-hand side is non-negative (if not, no solution exists).

Let me know if you'd like a graph or visual explanation!