Solve each equation or inequality. | 12- 6x | + 3 ≥ 9

Verified step by step guidance

1. Start by isolating the absolute value expression. Subtract 3 from both sides of the inequality: \(| 12 - 5x | + 3 \geq 9 \implies | 12 - 5x | \geq 6\).
2. Recall that for an absolute value inequality \(|A| \geq B\) (where \(B > 0\)), the solution splits into two cases: \(A \geq B\) or \(A \leq -B\).
3. Apply the two cases to the expression inside the absolute value: Case 1: \(12 - 5x \geq 6\) Case 2: \(12 - 5x \leq -6\).
4. Solve each inequality separately: For Case 1: Subtract 12 from both sides and then divide by -5, remembering to reverse the inequality sign when dividing by a negative number. For Case 2: Subtract 12 from both sides and then divide by -5, again reversing the inequality sign.
5. Write the solution as the union of the two solution sets found from Case 1 and Case 2. This represents all values of \(x\) that satisfy the original inequality.

Key Concepts

Absolute Value Inequalities

Absolute value inequalities involve expressions where the absolute value of a variable or expression is compared to a number. To solve them, you consider two cases: one where the expression inside the absolute value is positive or zero, and one where it is negative, leading to two separate inequalities.

Isolating the Absolute Value Expression

Before solving an absolute value inequality, isolate the absolute value term on one side of the inequality. This simplification allows you to apply the definition of absolute value inequalities correctly and split the problem into manageable cases.

Solving Linear Inequalities

After splitting the absolute value inequality into two linear inequalities, solve each by isolating the variable. This involves standard algebraic steps such as adding, subtracting, multiplying, or dividing both sides, while remembering to reverse the inequality sign when multiplying or dividing by a negative number.