Problem 35 - 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. $$ \frac{4-2 x}{3 x+4} \leq 0 $$

Step-by-Step Solution

Answer

The solution to the rational inequality is \((-∞,-4/3] \cup [2, ∞)\).

Step 1: Find the Critical Points

To find the critical points, set the numerator and denominator equal to zero and solve for x. Thus, \(4 - 2x = 0\) implies \(x = 2\), and \(3x + 4 = 0\) implies \(x = -4/3\) .

Step 2: Test each interval

The critical points divide the number line into 3 intervals: \(-∞ \rightarrow -4/3\), \(-4/3 \rightarrow 2\), and \(2 \rightarrow ∞\). Choose a test point from each interval and substitute it into the inequality. If the result is true, the interval is part of the solution set. If it's false, the interval is not part of the solution set.

Step 3: Determine the solution intervals

By testing the intervals, It can be determined that the inequality is satisfied for the intervals \(-∞ \rightarrow -4/3\) and \(2 \rightarrow ∞\).

Step 4: Write the solution in interval notation

The solution in interval notation is \((-∞,-4/3] \cup [2, ∞)\). The square brackets indicate that the endpoints are included in the solution because the original inequality contains equal to '≤'. If the inequality had been '<' or '>', then we'd use parentheses to indicate that the endpoints are not included.

Key Concepts

Critical Points

In the context of solving rational inequalities, critical points are of pivotal importance. They are specific values of the variable that can potentially change the sign of the expression. To find them, we set the numerator and the denominator of the rational expression to zero separately, because these are the values where the expression could be undefined or could change from positive to negative, or vice versa.

For instance, in the inequality \(\frac{4-2x}{3x+4} \leq 0\), we find that the critical points are \(x = 2\) and \(x = -\frac{4}{3}\), which we get from setting \(4-2x = 0\) and \(3x+4 = 0\) respectively. It's important to understand that these critical points divide the number line into intervals that we must then test to determine where the inequality holds true.

Interval Notation

Interval notation is a way to describe subsets of the real numbers, often used to express the solution sets of inequalities. It uses parentheses and brackets to denote if endpoints are excluded or included. For example, parentheses \((a, b)\) indicate that the endpoints \(a\) and \(b\) are not part of the interval, while square brackets \[a, b\] tell us that \(a\) and \(b\) are included.

In our rational inequality exercise, we expressed the solution \(\frac{4-2x}{3x+4} \leq 0\) with interval notation as \( (-\infty, -\frac{4}{3}] \cup [2, \infty) \). This means the solution includes the values from negative infinity up to and including \( -\frac{4}{3}\), and from \(2\) to positive infinity, without including the number \(2\) itself. This nuanced expression is compact yet rich with information, which makes interval notation both efficient and precise.

Inequality Graphing

Graphing inequalities on a number line offers a visual representation of the solution set. To graph the solution of a rational inequality, one plots the critical points and shades the intervals where the inequality is satisfied. In the case of \(\frac{4-2x}{3x+4} \leq 0\), we have two critical points at \(x = -\frac{4}{3}\) and \(x = 2\).

On a number line, these points divide the line into three intervals. We use test values from each interval to determine if they satisfy the inequality. The intervals where the inequality holds are then shaded. For example, the solution to our problem includes the intervals \( (-\infty, -\frac{4}{3}] \), represented by a darkened line leading up to and including the point \( -\frac{4}{3}\), and \( [2, \infty) \), indicated by shading starting just after \(2\) and continuing to infinity. Incorporating an inequality graph in the solving process not only aids understanding but also serves as a tool for checking the accuracy of the solution.