Question

A student attempted to solve the inequality \(\frac{x+4}{x-3} \leq 0\) by multiplying both sides of the inequality by \(x-3\) to get \(x+4 \leq 0\). This led to a solution of \(\\{x \mid x \leq-4\\}\). Is the student correct? Explain.

Short answer

The correct solution set is \(-4 \leq x < 3\). The student made an error by not considering the sign change of \(x-3\).

Step-by-step solution

- Analyze the inequality

The original inequality is \( \frac{x+4}{x-3} \leq 0 \). Our goal is to determine the values of \(x\) for which the inequality holds.

- Understand the critical points

The numerator \(x+4\) can be zero when \(x = -4\). The denominator \(x-3\) is zero when \(x = 3\). These points divide the number line into intervals.

- Determine the sign of each interval

Test points from each interval: - For \(x < -4\), select \(x = -5\): \( \frac{-5+4}{-5-3} = \frac{-1}{-8} = \frac{1}{8} > 0 \)- For \(-4 < x < 3\), select \(x = 0\): \( \frac{0+4}{0-3} = \frac{4}{-3} < 0 \)- For \(x > 3\), select \(x = 4\): \( \frac{4+4}{4-3} = \frac{8}{1} = 8 > 0 \)

- Consider equality and undefined points

The inequality \( \frac{x+4}{x-3} \leq 0 \) is valid where the fraction is zero or negative. Include \(x = -4\) since \( \frac{-4+4}{-4-3} = 0 \), but exclude \(x = 3\) as the fraction is undefined at that point.

- Compile the solution

Based on the intervals and inclusion of necessary points, the solution set is \( -4 \leq x < 3 \).

- Analyze the student's mistake

The student incorrectly assumed \(x-3\) is always positive; this led to a wrong interval. They should have considered changing signs depending on the intervals.

Key concepts

Inequality

When solving inequalities, you don’t just find solutions that make an equation true. Instead, you identify a range of values that make the inequality true. For example, with \(\frac{x+4}{x-3} \leq 0\), we need to find where the rational expression \( \frac{x+4}{x-3} \) is less than or equal to zero. This means both the numerator and denominator need to be considered. Inequality problems often need sign analysis and can involve critical points where the expression changes its behavior.

Unlike equations, inequalities involve testing intervals to ensure a solution makes the entire inequality true. Knowing these intervals depend on where the components of the inequality equal zero or become undefined.

Critical Points

Critical points in an inequality are where the expression either equals zero or is undefined. For the inequality \( \frac{x+4}{x-3} \leq 0\), the critical points arise from:

• \text{x= -4, making the numerator 0}
• \text{x = 3, making the denominator 0}

These critical points divide the number line into different intervals that need to be checked separately. Understanding these values is fundamental as they determine where the inequality can change its state from positive to negative or vice versa.

Sign Analysis

Sign analysis is the process of determining if an expression is positive or negative within each interval identified by critical points. For \[ \frac{x+4}{x-3} \] , we break the number line into three intervals, based on the critical points (-4 and 3). We then test points within these intervals to determine the sign:

• \textbf{-∞ to -4:} Pick x = -5, yielding \[ \frac{-5+4}{-5-3} = \frac{-1}{-8} = \frac{1}{8} \gt 0 \]
• \textbf{-4 to 3:} Pick x = 0, yielding \[ \frac{0+4}{0-3} = \frac{4}{-3} \lt 0 \]
• \textbf{3 to ∞:} Pick x = 4, yielding \[ \frac{4+4}{4-3} = 8 \gt 0 \]

This sign analysis reveals that the original inequality \[ \frac{x+4}{x-3} \leq 0 \] is only true in the interval where the expression is zero or negative, i.e., when -4 \leq x < 3.

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. Solving inequalities involving rational expressions requires understanding their properties, especially when the denominator creates undefined points. The inequality in question, \[ \frac{x+4}{x-3} \leq 0 \], demonstrates this concept:
\text{* The numerator x+4 becomes zero at x = -4}
\text{* The denominator x-3 becomes zero at x = 3}
These points must be excluded or included appropriately in the solution based on their respective effects on the inequality. In our case, \[ x = -4 \] is included because the inequality \[ \frac{-4+4}{-4-3} = 0 \] holds true. Conversely, \[ x = 3 \] is excluded as it makes the expression undefined. Therefore, the solution to the inequality \[ \frac{x+4}{x-3} \leq 0 \] is the interval -4 \leq x < 3.