https://file.aiquickdraw.com/mathbot/file/17258765797055u61sdtb.jpg

The inequality shown in the image is:

\[
\frac{2x + 1}{x - 9} \leq 0
\]

To solve this rational inequality, we need to follow a step-by-step process:

### Step 1: Identify the critical points
The critical points are values of \(x\) where the numerator or denominator is zero. These points help us divide the number line into intervals to test.

1. **Numerator**: Set \(2x + 1 = 0\) to find the zero of the numerator.
   \[
   2x + 1 = 0 \quad \Rightarrow \quad 2x = -1 \quad \Rightarrow \quad x = -\frac{1}{2}
   \]
   So, one critical point is \(x = -\frac{1}{2}\).

2. **Denominator**: Set \(x - 9 = 0\) to find where the denominator is zero.
   \[
   x - 9 = 0 \quad \Rightarrow \quad x = 9
   \]
   So, another critical point is \(x = 9\), but we need to exclude this point from the solution, as division by zero is undefined.

### Step 2: Determine the sign of the expression in each interval
We will check the sign of \(\frac{2x + 1}{x - 9}\) in the intervals determined by the critical points: \((- \infty, -\frac{1}{2})\), \((- \frac{1}{2}, 9)\), and \((9, \infty)\).

1. **Interval \((- \infty, -\frac{1}{2})\)**:
   - Choose a test point, say \(x = -1\):
     \[
     \frac{2(-1) + 1}{-1 - 9} = \frac{-2 + 1}{-10} = \frac{-1}{-10} = \frac{1}{10} > 0
     \]
     So, the expression is positive in this interval.

2. **Interval \((- \frac{1}{2}, 9)\)**:
   - Choose a test point, say \(x = 0\):
     \[
     \frac{2(0) + 1}{0 - 9} = \frac{1}{-9} = -\frac{1}{9} < 0
     \]
     So, the expression is negative in this interval.

3. **Interval \((9, \infty)\)**:
   - Choose a test point, say \(x = 10\):
     \[
     \frac{2(10) + 1}{10 - 9} = \frac{20 + 1}{1} = 21 > 0
     \]
     So, the expression is positive in this interval.

### Step 3: Include the critical points where the inequality holds
We want the expression to be **less than or equal to zero** (\(\leq 0\)). This means we include points where the expression is negative or zero.

- The expression is negative in the interval \((- \frac{1}{2}, 9)\).
- At \(x = -\frac{1}{2}\), the expression is zero, so we include this point.
- At \(x = 9\), the expression is undefined, so we exclude this point.

### Final Solution:
The solution to the inequality is:

\[
x \in \left[-\frac{1}{2}, 9\right)
\]

Would you like further details or have any other questions?

Here are 5 related questions to expand on this:
1. How would the solution change if the inequality was strict (\(<\) instead of \(\leq\))?
2. Can you solve similar inequalities with more complex numerators or denominators?
3. How do we handle cases where the denominator has more than one factor?
4. What happens if the inequality is a rational function greater than zero?
5. Can you graph the solution of this inequality on a number line?

**Tip:** Always check for critical points in both the numerator and the denominator when solving rational inequalities.

Solve the inequality \(\frac{2x + 1}{x - 9} \leq 0\).

Learn how to solve the rational inequality \(\frac{2x + 1}{x - 9} \leq 0\) by finding critical points, dividing the number line into intervals, and determining the sign of the expression in each interval. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation