https://file.aiquickdraw.com/mathbot/file/17255675724336qf9kqom.jpg

The inequality in the image is:

\[
\frac{2x + 10}{x + 3} \geq 1
\]

Let's solve this inequality step-by-step.

### Step 1: Simplify the Inequality
Subtract 1 from both sides to get a single fraction:

\[
\frac{2x + 10}{x + 3} - 1 \geq 0
\]

Simplify the left side:

\[
\frac{2x + 10 - (x + 3)}{x + 3} \geq 0
\]

\[
\frac{2x + 10 - x - 3}{x + 3} \geq 0
\]

\[
\frac{x + 7}{x + 3} \geq 0
\]

### Step 2: Find Critical Points
The critical points occur where the numerator and the denominator are zero:

- Numerator: \(x + 7 = 0 \implies x = -7\)
- Denominator: \(x + 3 = 0 \implies x = -3\)

These points divide the number line into intervals.

### Step 3: Test Intervals
The critical points divide the real line into three intervals: \(x < -7\), \(-7 < x < -3\), and \(x > -3\).

We need to test these intervals to see where the expression \(\frac{x + 7}{x + 3}\) is greater than or equal to zero.

1. **For \(x < -7\):** 
   - Choose \(x = -8\): \(\frac{-8 + 7}{-8 + 3} = \frac{-1}{-5} = \frac{1}{5} > 0\).
2. **For \(-7 < x < -3\):** 
   - Choose \(x = -5\): \(\frac{-5 + 7}{-5 + 3} = \frac{2}{-2} = -1 < 0\).
3. **For \(x > -3\):**
   - Choose \(x = 0\): \(\frac{0 + 7}{0 + 3} = \frac{7}{3} > 0\).

### Step 4: Include Critical Points
Now check whether to include \(x = -7\) and \(x = -3\):

- At \(x = -7\), \(\frac{x + 7}{x + 3} = 0\), which satisfies the inequality.
- At \(x = -3\), the expression is undefined.

### Step 5: Write the Solution
The solution to the inequality is:

\[
x \in (-\infty, -7] \cup (-3, \infty)
\]

Would you like any further details or have any questions? Here are some related questions to explore:

1. What happens when the denominator has other critical points?
2. How do we handle inequalities with absolute values?
3. What changes if the inequality sign is reversed?
4. How do you solve inequalities with quadratic expressions in the numerator?
5. How can graphing help in solving rational inequalities?

**Tip:** Always remember to check critical points where the function is undefined, as these can change the intervals of your solution!

Solve the inequality: (2x + 10) / (x + 3) ≥ 1

Explore the step-by-step solution for the inequality (2x + 10) / (x + 3) ≥ 1. This problem teaches rational expressions, critical points, and testing intervals. Suitable for students in Grades 10-12.

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