In Exercises 61–76, solve each absolute value equation or indicate that the equation has no solution. 7|5x| + 2 = 16

Verified step by step guidance

1. Start by isolating the absolute value expression. Subtract 2 from both sides of the equation: \(7|5x| + 2 - 2 = 16 - 2\), which simplifies to \(7|5x| = 14\).
2. Next, divide both sides of the equation by 7 to solve for \(|5x|\): \(\frac{7|5x|}{7} = \frac{14}{7}\), giving \(|5x| = 2\).
3. Recall that the absolute value equation \(|A| = B\) means \(A = B\) or \(A = -B\). So, set up two equations: \(5x = 2\) and \(5x = -2\).
4. Solve each equation for \(x\) by dividing both sides by 5: \(x = \frac{2}{5}\) and \(x = \frac{-2}{5}\).
5. These two values are the solutions to the original equation. You can check each by substituting back into the original equation to verify correctness.

Key Concepts

Absolute Value Definition

The absolute value of a number represents its distance from zero on the number line, always yielding a non-negative result. For any real number x, |x| equals x if x is positive or zero, and -x if x is negative. Understanding this helps in setting up equations when absolute values are involved.

Solving Absolute Value Equations

To solve an equation involving absolute values, isolate the absolute value expression first. Then, split the equation into two cases: one where the expression inside the absolute value equals the positive value, and one where it equals the negative value. Solve both cases separately to find all possible solutions.

Checking for Extraneous Solutions

After solving absolute value equations, it is important to verify each solution by substituting back into the original equation. Some solutions may not satisfy the original equation due to the nature of absolute values, so checking prevents including invalid answers.