Solve each rational inequality. Give the solution set in interval notation. See Examples 8 and 9. 7/(x+2)≥1/(x+2)

Verified step by step guidance

1. Start by writing down the inequality: \(\frac{7}{x+2} \geq \frac{1}{x+2}\).
2. Identify the critical values by setting the denominators equal to zero: solve \(x + 2 = 0\) which gives \(x = -2\). This value is excluded from the domain because it makes the denominator zero.
3. Since the denominators are the same and not zero (except at \(x = -2\)), subtract \(\frac{1}{x+2}\) from both sides to get a single rational expression: \(\frac{7}{x+2} - \frac{1}{x+2} \geq 0\).
4. Combine the fractions: \(\frac{7 - 1}{x+2} \geq 0\), which simplifies to \(\frac{6}{x+2} \geq 0\).
5. Analyze the inequality \(\frac{6}{x+2} \geq 0\): since the numerator 6 is positive, the sign of the expression depends on the denominator \(x+2\). Determine where \(x+2\) is positive or negative, and write the solution set accordingly, excluding \(x = -2\).

Key Concepts

Rational Inequalities

Rational inequalities involve expressions with variables in the denominator. Solving them requires finding values of the variable that make the inequality true, while considering restrictions where the denominator is zero to avoid undefined expressions.

Domain Restrictions

The domain of a rational expression excludes values that make the denominator zero. Identifying these restrictions is crucial before solving inequalities, as these points cannot be included in the solution set and often split the number line into test intervals.

Interval Notation and Test Intervals

After determining critical points from the inequality and domain restrictions, the number line is divided into intervals. Testing values from each interval helps determine where the inequality holds true. Solutions are then expressed using interval notation to clearly represent all valid values.