Questions: Solve the inequality, and graph the solution set. (4x+5)/(x-2) ≤ 0

Solve the inequality, and graph the solution set. (4x+5)/(x-2) ≤ 0
Transcript text: Solve the inequality, and graph the solution set. \[ \frac{4 x+5}{x-2} \leq 0 \]
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Solution

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Solution Steps

To solve the inequality \(\frac{4x+5}{x-2} \leq 0\), we need to find the values of \(x\) that make the expression non-positive. This involves finding the critical points where the expression is zero or undefined, and then testing intervals between these points to determine where the inequality holds.

  1. Find Critical Points: Set the numerator equal to zero to find where the expression is zero, and set the denominator equal to zero to find where the expression is undefined.
  2. Test Intervals: Use the critical points to divide the number line into intervals. Test a point from each interval to see if the inequality holds.
  3. Combine Results: Determine the solution set based on the intervals where the inequality is satisfied.
Step 1: Identify Critical Points

To solve the inequality

\[ \frac{4x + 5}{x - 2} \leq 0 \]

we first identify the critical points where the expression is either zero or undefined.

  1. Zero of the numerator: Set the numerator equal to zero. \[ 4x + 5 = 0 \] Solving for \(x\), \[ 4x = -5 \quad \Rightarrow \quad x = -\frac{5}{4} \]

  2. Undefined point: Set the denominator equal to zero. \[ x - 2 = 0 \] Solving for \(x\), \[ x = 2 \]

The critical points are \(x = -\frac{5}{4}\) and \(x = 2\).

Step 2: Test Intervals

The critical points divide the number line into three intervals: \((- \infty, -\frac{5}{4})\), \((- \frac{5}{4}, 2)\), and \((2, \infty)\). We will test each interval to determine where the inequality holds.

  1. Interval \((- \infty, -\frac{5}{4})\): Choose \(x = -2\). \[ \frac{4(-2) + 5}{-2 - 2} = \frac{-8 + 5}{-4} = \frac{-3}{-4} = \frac{3}{4} > 0 \] The inequality does not hold in this interval.

  2. Interval \((- \frac{5}{4}, 2)\): Choose \(x = 0\). \[ \frac{4(0) + 5}{0 - 2} = \frac{5}{-2} = -\frac{5}{2} < 0 \] The inequality holds in this interval.

  3. Interval \((2, \infty)\): Choose \(x = 3\). \[ \frac{4(3) + 5}{3 - 2} = \frac{12 + 5}{1} = 17 > 0 \] The inequality does not hold in this interval.

Step 3: Include Critical Points
  • At \(x = -\frac{5}{4}\), the expression is zero, which satisfies the \(\leq 0\) condition.
  • At \(x = 2\), the expression is undefined, so it cannot be included in the solution set.

Final Answer

The solution to the inequality is:

\[ x \in \left[-\frac{5}{4}, 2\right) \]

Thus, the solution set is:

\[ \boxed{x \in \left[-\frac{5}{4}, 2\right)} \]

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