Solve each equation or inequality. | 10- 4x | ≥ -4

Verified step by step guidance

1. Recognize that the inequality involves an absolute value expression: \(|10 - 4x| \geq 4\). Recall that for any expression \(A\), the inequality \(|A| \geq k\) (where \(k > 0\)) means \(A \leq -k\) or \(A \geq k\).
2. Set up two separate inequalities based on the definition of absolute value inequalities: 1) \(10 - 4x \geq 4\) 2) \(10 - 4x \leq -4\)
3. Solve the first inequality \(10 - 4x \geq 4\) by isolating \(x\): Subtract 10 from both sides: \(-4x \geq 4 - 10\) Simplify the right side: \(-4x \geq -6\) Divide both sides by \(-4\), remembering to reverse the inequality sign because you are dividing by a negative number: \(x \leq \frac{-6}{-4}\)
4. Solve the second inequality \(10 - 4x \leq -4\) similarly: Subtract 10 from both sides: \(-4x \leq -4 - 10\) Simplify the right side: \(-4x \leq -14\) Divide both sides by \(-4\), reversing the inequality sign: \(x \geq \frac{-14}{-4}\)
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 absolute value of a variable or expression is compared to a number. To solve |A| ≥ B, where B ≥ 0, the solution splits into two cases: A ≥ B or A ≤ -B. This approach helps handle the distance interpretation of absolute values.

Solving Linear Inequalities

Linear inequalities involve expressions with variables to the first power and inequality signs. Solving them requires isolating the variable on one side, while remembering to reverse the inequality sign when multiplying or dividing by a negative number. Solutions are often expressed as intervals or inequalities.

Properties of Inequalities

Understanding how inequalities behave under addition, subtraction, multiplication, and division is crucial. For example, multiplying or dividing both sides by a negative number reverses the inequality sign. These properties ensure correct manipulation of inequalities during problem solving.