Solve each equation or inequality. | 7 + 2x| = 0

Verified step by step guidance

1. Recognize that the equation involves an absolute value expression: \(|7 + 2x| = 0\).
2. Recall that the absolute value of a number is always non-negative, and it equals zero only when the expression inside the absolute value is zero.
3. Set the expression inside the absolute value equal to zero: \(7 + 2x = 0\).
4. Solve the linear equation for \(x\) by isolating \(x\): subtract 7 from both sides to get \(2x = -7\), then divide both sides by 2 to find \(x = \frac{-7}{2}\).
5. Conclude that the solution to the equation \(|7 + 2x| = 0\) is the value of \(x\) found in the previous step.

Key Concepts

Absolute Value Definition

The absolute value of a number represents its distance from zero on the number line and is always non-negative. For any expression |A|, the result is zero only if A itself equals zero. This property is crucial when solving equations involving absolute values.

Solving Absolute Value Equations

To solve an equation like |A| = B, where B ≥ 0, set A equal to B and also to -B, then solve both resulting equations. If B is negative, the equation has no solution because absolute values cannot be negative.

Linear Equations

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. Solving linear equations involves isolating the variable to find its value, which is essential after removing the absolute value.