Problem 34 - College Algebra

Question

Solve each rational inequality in Exercises \(29-48,\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ \frac{-x-3}{x+2} \leq 0 $$

Step-by-Step Solution

Answer

The solution to the rational inequality is \((-3, -2] ∪ [-2, ∞)\) in interval notation.

Step 1: Factoring and Critical Values

Factor the expression on the left-hand side of the inequality if it is factorable and identify the critical values. For the inequality \(-\frac{x+3}{x+2} \leq 0\), the critical values are -3 from the numerator and -2 from the denominator.

Step 2: Testing Intervals

Split the real number line into intervals using the critical values, and choose a test point within each interval. Testing is done by substitution of the test points into the factored form of the rational expression. The intervals are \((-∞,-3)\), \((-3,-2)\), and \((-2,∞)\). Select -4, -2.5, and 0 as test points.

Step 3: Solving the Inequality

Substituting test points -4, -2.5, and 0 into the factored form of the rational expression tells us if the inequality is less than or equal to zero. Since \(-\frac{x+3}{x+2} ≤ 0\), the two intervals that satisfy the inequality are \((-3,-2)\) and \((-2, ∞)\).

Step 4: Interval Notation

In interval notation, the solution set is represented as \((-3, -2] ∪ [-2, ∞)\). This represents all x values that make the inequality true.

Key Concepts

Rational Expressions

A rational expression is a fraction that has polynomials in both its numerator and denominator. Solving inequalities involving rational expressions requires a step-by-step approach. The first step usually involves factoring both the numerator and denominator, if possible. In these inequalities, we must be aware that division by zero is undefined, which affects the values that 'x' can take.

In an inequality such as \(\frac{-x-3}{x+2} \leq 0\), both the numerator and the denominator can influence where the expression is positive or negative. For instance, when the numerator is zero, the whole expression is zero, and if the denominator is zero, the expression is undefined. This behavior is key in finding the solution to the inequality.

Interval Notation

When presenting solutions to inequalities, especially those involving rational expressions, interval notation is used for its conciseness and clarity. Interval notation allows us to express a range of numbers, where the smallest number is listed first, followed by the largest.

An open interval, such as \( (-3, -2) \), indicates that the 'x' values are more than -3 and less than -2, excluding the endpoints. Meanwhile, a closed interval like \( [-2, \infty) \), includes the endpoint -2, indicating that the value -2 satisfies the inequality, and continues indefinitely towards positive infinity.

Critical Values

The concept of critical values is integral when graphing rational inequalities. Critical values are the x-values where the numerator is zero (providing possible zeros of the whole expression) or where the denominator is zero (producing vertical asymptotes since the expression becomes undefined).

For the inequality \(\frac{-x-3}{x+2} \leq 0\), we identify -3 and -2 as critical values because \( -x-3 = 0 \) at \( x=-3 \) and \( x+2 = 0 \) at \( x=-2 \). These points are essential in establishing intervals to be tested for the inequality and are pivotal in sketching the graph.

Inequality Graphing

Graphing an inequality is a visual representation which shows where a rational expression is less than, greater than, equal to, or not equal to zero on a number line. It prominently features the critical values as boundaries between intervals.

After testing intervals formed by critical values with test points, we determine where the inequality holds true. For example, with \(\frac{-x-3}{x+2} \leq 0\), after testing we see the inequality is satisfied in intervals where the test point yields a negative result. We mark these intervals on a number line, often with a solid dot for included endpoints (indicating 'or equal to') and an open circle for excluded endpoints. This step visualizes the solution set and complements the interval notation.