Solve each equation or inequality.
$\mid 8-5x\mid \ge 2$

Verified step by step guidance

1. Recognize that the inequality involves an absolute value expression: \(|8 - 5x| \geq 2\). Recall that for any expression \(A\), \(|A| \geq k\) means \(A \geq k\) or \(A \leq -k\) when \(k > 0\).
2. Set up two separate inequalities based on the definition of absolute value inequalities: 1) \(8 - 5x \geq 2\) 2) \(8 - 5x \leq -2\)
3. Solve the first inequality \(8 - 5x \geq 2\) by isolating \(x\): Subtract 8 from both sides: \(-5x \geq 2 - 8\) Simplify the right side: \(-5x \geq -6\) Divide both sides by \(-5\), remembering to reverse the inequality sign because you are dividing by a negative number: \(x \leq \frac{-6}{-5}\)
4. Solve the second inequality \(8 - 5x \leq -2\) similarly: Subtract 8 from both sides: \(-5x \leq -2 - 8\) Simplify the right side: \(-5x \leq -10\) Divide both sides by \(-5\), reversing the inequality sign: \(x \geq \frac{-10}{-5}\)
5. Combine the two solution sets from steps 3 and 4 to express the final solution as a union of intervals where \(x\) satisfies either inequality.

Key Concepts

Absolute Value Inequalities

Absolute value inequalities involve expressions where the distance of a number from zero is compared to another value. For |A| ≥ B, the solution includes values where A ≤ -B or A ≥ B, reflecting the two-sided nature of absolute value.

Solving Linear Inequalities

Solving linear inequalities requires isolating the variable while maintaining inequality direction. When multiplying or dividing by a negative number, the inequality sign must be reversed to preserve the inequality's truth.

Splitting Absolute Value Expressions

To solve |expression| ≥ number, split the inequality into two cases: one where the expression is greater than or equal to the number, and one where it is less than or equal to the negative of that number. Solve each case separately.