In Exercises 59–94, solve each absolute value inequality. 5|2x + 1| - 3 ≥ 9

Verified step by step guidance

1. Start by isolating the absolute value expression. Add 3 to both sides of the inequality: $5|2x+1|-3+3\ge 9+3$.
2. Simplify the inequality: $5|2x+1|\ge 12$.
3. Divide both sides by 5 to further isolate the absolute value: $|2x+1|\ge \frac{12}{5}$.
4. Recall that an absolute value inequality of the form $|A|\ge B$ can be split into two separate inequalities: $A\ge B$ or $A\le -B$.
5. Apply this to the expression: $2x+1\ge \frac{12}{5}$ or $2x+1\le -\frac{12}{5}$. Solve each inequality separately for $x$.

Key Concepts

Absolute Value

Absolute value represents 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 inequalities that involve expressions within absolute value symbols.

Inequalities

Inequalities express a relationship between two expressions that are not necessarily equal. They can be represented using symbols such as >, <, ≥, or ≤. When solving inequalities, especially those involving absolute values, it is important to consider the different cases that arise from the definition of absolute value, leading to multiple potential solutions.

Solving Absolute Value Inequalities

To solve an absolute value inequality, one must isolate the absolute value expression and then split the inequality into two separate cases. For example, if |A| ≥ B, it translates to A ≥ B or A ≤ -B. This method allows for finding all possible solutions that satisfy the original inequality, which is essential for complete problem-solving.