Solve each inequality. Give the solution set using interval notation. See Examples 8 and 9. 4x + 7 ———— ≤ 2x + 5 -3

Verified step by step guidance

1. Start by rewriting the inequality clearly: \(\frac{4x + 7}{-3} \leq 2x + 5\).
2. Multiply both sides of the inequality by \(-3\) to eliminate the denominator. Remember, multiplying by a negative number reverses the inequality sign, so the inequality becomes: \(4x + 7 \geq -3(2x + 5)\).
3. Distribute the \(-3\) on the right side: \(4x + 7 \geq -6x - 15\).
4. Collect like terms by adding \$6x\( to both sides and subtracting \)7$ from both sides: \(4x + 6x \geq -15 - 7\), which simplifies to \(10x \geq -22\).
5. Finally, divide both sides by \$10$ (a positive number, so the inequality sign stays the same): \(x \geq \frac{-22}{10}\). Express the solution set in interval notation as \([\frac{-22}{10}, \infty)\).

Key Concepts

Solving Rational Inequalities

Rational inequalities involve expressions with variables in the numerator and denominator. To solve them, first bring all terms to one side to form a single rational expression, then determine where the expression is positive or negative by analyzing critical points from the numerator and denominator.

Critical Points and Sign Analysis

Critical points occur where the numerator or denominator equals zero, dividing the number line into intervals. By testing values in each interval, you can determine the sign of the rational expression, which helps identify where the inequality holds true.

Interval Notation and Domain Restrictions

Interval notation expresses solution sets compactly using parentheses and brackets. When solving inequalities with denominators, exclude values that make the denominator zero, as these are not in the domain, ensuring the solution set respects these restrictions.