Solve each inequality. Give the solution set using interval notation. (x+7) / (2x+1) ≤1

Verified step by step guidance

1. Start by writing the inequality clearly: \(\frac{x+7}{2x+1} \leq 1\).
2. Bring all terms to one side to have zero on the other side: \(\frac{x+7}{2x+1} - 1 \leq 0\).
3. Combine the terms over a common denominator: \(\frac{x+7 - (2x+1)}{2x+1} \leq 0\).
4. Simplify the numerator: \(\frac{x+7 - 2x - 1}{2x+1} = \frac{-x + 6}{2x+1} \leq 0\).
5. Determine the critical points by setting numerator and denominator equal to zero separately: solve \(-x + 6 = 0\) and \$2x + 1 = 0$, then analyze the sign of the expression \(\frac{-x + 6}{2x+1}\) on intervals defined by these points to find where it is less than or equal to zero.

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 compare against zero, then find critical points where the numerator or denominator is zero. These points divide the number line into intervals to test for solution validity.

Critical Points and Sign Analysis

Critical points occur where the numerator or denominator equals zero, as these values can change the inequality's truth. By testing values from each interval created by these points, you determine where the inequality holds true. Remember to exclude points that make the denominator zero.

Interval Notation

Interval notation expresses solution sets as ranges of values. Use parentheses () to exclude endpoints and brackets [] to include them. For inequalities involving ≤ or ≥, include points where the expression equals the boundary value, unless undefined.