Solve each inequality. Give the solution set in interval notation. (2x-5)/-8≤1-x

Verified step by step guidance

1. Start with the given inequality: \(\frac{2x - 5}{-8} \leq 1 - x\).
2. To eliminate the fraction, multiply both sides of the inequality by \(-8\). Remember, when multiplying or dividing an inequality by a negative number, you must reverse the inequality sign. So, multiplying both sides by \(-8\) gives: \(2x - 5 \geq -8(1 - x)\).
3. Distribute \(-8\) on the right side: \(2x - 5 \geq -8 + 8x\).
4. Next, get all terms involving \(x\) on one side and constants on the other. Subtract \$2x$ from both sides: \(-5 \geq -8 + 6x\). Then add \(8\) to both sides: \(3 \geq 6x\).
5. Finally, divide both sides by \(6\) (a positive number, so inequality direction stays the same): \(\frac{3}{6} \geq x\), which simplifies to \(\frac{1}{2} \geq x\). This means \(x \leq \frac{1}{2}\). Express the solution set in interval notation as \((-\infty, \frac{1}{2}]\).

Key Concepts

Solving Linear Inequalities

Solving linear inequalities involves isolating the variable on one side to find the range of values that satisfy the inequality. Similar to equations, operations like addition, subtraction, multiplication, and division are used, but special care is needed when multiplying or dividing by negative numbers, as this reverses the inequality sign.

Handling Inequalities with Fractions

When inequalities contain fractions, it is often helpful to clear denominators by multiplying both sides by the least common denominator. This simplifies the inequality to a linear form, but remember to consider the sign of the denominator when multiplying or dividing, as it affects the inequality direction.

Interval Notation for Solution Sets

Interval notation is a concise way to express the set of all solutions to an inequality. It uses parentheses for values not included (open intervals) and brackets for values included (closed intervals). For example, [a, b) means all values from a to b, including a but excluding b.