Solve each equation or inequality. | 12- 5x | + 3 ≥ 9

Verified step by step guidance

1. Start by isolating the absolute value expression. Subtract 3 from both sides of the inequality: \(|12 - 5x| + 3 - 3 \geq 9 - 3\).
2. This simplifies to \(|12 - 5x| \geq 6\).
3. To solve the inequality \(|12 - 5x| \geq 6\), consider the two cases for the absolute value: \(12 - 5x \geq 6\) and \(12 - 5x \leq -6\).
4. For the first case, solve \(12 - 5x \geq 6\) by subtracting 12 from both sides: \(-5x \geq -6\). Then, divide by -5, remembering to reverse the inequality sign: \(x \leq \frac{6}{5}\).
5. For the second case, solve \(12 - 5x \leq -6\) by subtracting 12 from both sides: \(-5x \leq -18\). Then, divide by -5, remembering to reverse the inequality sign: \(x \geq \frac{18}{5}\).

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 expressions within absolute value symbols.

Inequalities

Inequalities express a relationship between two expressions that are not necessarily equal. They use symbols such as ≥ (greater than or equal to) and ≤ (less than or equal to) to indicate the nature of the relationship. Solving inequalities often involves similar steps to solving equations, but requires careful consideration of the direction of the inequality when multiplying or dividing by negative numbers.

Solving Equations

Solving equations involves finding the value(s) of the variable that make the equation true. This process often includes isolating the variable on one side of the equation through various algebraic manipulations, such as adding, subtracting, multiplying, or dividing both sides. In the context of absolute value equations or inequalities, it may require considering multiple cases based on the definition of absolute value.