Solve each equation or inequality. |7x+8| - 6 > -3

Verified step by step guidance

1. Start by isolating the absolute value expression. Add 6 to both sides of the inequality: $|7x+8|-6+6>-3+6$.
2. This simplifies to $|7x+8|>3$.
3. To solve the inequality $|7x+8|>3$, consider the two cases for absolute value inequalities: $7x+8>3$ and $7x+8<-3$.
4. For the first case, solve $7x+8>3$ by subtracting 8 from both sides: $7x>3-8$.
5. For the second case, solve $7x+8<-3$ by subtracting 8 from both sides: $7x<-3-8$.

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 x, the absolute value is denoted as |x| and is defined as |x| = x if x ≥ 0, and |x| = -x if x < 0. Understanding absolute value is crucial for solving equations and inequalities that involve it, as it can lead to two separate cases to consider.

Inequalities

An inequality is a mathematical statement that compares two expressions, indicating that one is greater than, less than, or not equal to the other. In this case, the inequality |7x + 8| - 6 > -3 requires manipulation to isolate the variable. Solving inequalities often involves similar steps to solving equations, but special attention must be paid to the direction of the inequality when multiplying or dividing by negative numbers.

Case Analysis

When dealing with absolute value equations or inequalities, case analysis is a method used to break down the problem into simpler parts. For the expression |7x + 8|, we consider two scenarios: when the expression inside the absolute value is non-negative (7x + 8 ≥ 0) and when it is negative (7x + 8 < 0). This approach allows us to solve for x in each case separately, leading to a complete solution for the original inequality.