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Build your own math worksheets from 21,000 problems for grades 3 to 12, from fractions to calculus. Every problem includes step-by-step solutions.

Absolute value equations

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5349619
Use the graphs of \(f(x)=|x-2|\) and \(g(x)=1\) to solve \(|x-2|=1\).
Figure for problem 534961

Hints

- The absolute value \(|x-2|\) represents the distance from \(x\) to \(2\). - Find where the V-shaped graph has height \(1\).

Solution

1. The graph of \(f(x)=|x-2|\) is a V with vertex \((2, 0)\). 2. The graph of \(g(x)=1\) is a horizontal line. 3. The graphs intersect at \(x=1\) and \(x=3\), so these are the solutions.

Answer

\(x=1\) or \(x=3\)
5143409
Find all real numbers \(x\) that satisfy \(\sqrt{(x-4)^2}=9\). Justify your solution using the relationship among square roots, squares, and absolute value.

Hints

- Rewrite the square root of a square as an absolute value. - An expression with absolute value \(9\) can equal either \(9\) or \(-9\). - Check both solutions in the original equation.

Solution

1. Use \(\sqrt{u^2}=|u|\) to rewrite the equation as \(|x-4|=9\). 2. Therefore, \(x-4=9\) or \(x-4=-9\). 3. Solving gives \(x=13\) or \(x=-5\). 4. Both values check in the original equation.

Answer

\(x=13\) or \(x=-5\)

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