In Exercises 101–106, solve each equation. |√x - 5| = 2

Verified step by step guidance

1. Recognize that the equation involves an absolute value: \(|\sqrt{x} - 5| = 2\). This means the expression inside the absolute value can be either positive or negative.
2. Set up two separate equations to account for both scenarios: \(\sqrt{x} - 5 = 2\) and \(\sqrt{x} - 5 = -2\).
3. Solve the first equation: \(\sqrt{x} - 5 = 2\). Add 5 to both sides to isolate the square root: \(\sqrt{x} = 7\).
4. Solve the second equation: \(\sqrt{x} - 5 = -2\). Add 5 to both sides to isolate the square root: \(\sqrt{x} = 3\).
5. Square both sides of each equation to solve for \(x\): For \(\sqrt{x} = 7\), square to get \(x = 49\). For \(\sqrt{x} = 3\), square to get \(x = 9\).

Key Concepts

Absolute Value

Absolute value measures the distance of a number from zero on the number line, regardless of direction. For any real number 'a', the absolute value is defined as |a| = a if a ≥ 0, and |a| = -a if a < 0. In the context of equations, this means that |x| = k can lead to two possible equations: x = k and x = -k.

Solving Absolute Value Equations

To solve an equation involving absolute values, such as |√x - 5| = 2, we must consider both cases that arise from the definition of absolute value. This results in two separate equations: √x - 5 = 2 and √x - 5 = -2. Each equation can then be solved independently to find the possible values of x.

Square Roots and Squaring

When dealing with square roots, it is important to remember that squaring both sides of an equation can eliminate the square root but may introduce extraneous solutions. For example, if we have √x = a, squaring both sides gives x = a². Care must be taken to check all solutions in the original equation to ensure they are valid.