2. [-/2 Points] DETAILS MY NOTES LARPCALC10 2.7.043. \( \begin{array}{l}\text { Solve the inequality. (Enter your answer using interval notation.) } \\ \frac{3 x+7}{x-1}<2\end{array} \) Graph the solution set.

To solve the inequality: \[ \frac{3x + 7}{x - 1} < 2 \] ### Step 1: Find Critical Points First, determine where the expression is equal to 2 and where it's undefined. 1. **Set the expression equal to 2:** \[ \frac{3x + 7}{x - 1} = 2 \] Multiply both sides by \( x - 1 \) (noting that \( x \neq 1 \)): \[ 3x + 7 = 2(x - 1) \] \[ 3x + 7 = 2x - 2 \] \[ x = -9 \] 2. **Find where the denominator is zero:** \[ x - 1 = 0 \Rightarrow x = 1 \] So, the critical points are \( x = -9 \) and \( x = 1 \). ### Step 2: Test Intervals Divide the number line into intervals based on the critical points and test a value from each interval in the original inequality. 1. **Interval \((-∞, -9)\):** Choose \( x = -10 \): \[ \frac{3(-10) + 7}{-10 - 1} = \frac{-30 + 7}{-11} = \frac{-23}{-11} \approx 2.09 > 2 \quad \text{(Not in solution)} \] 2. **Interval \((-9, 1)\):** Choose \( x = 0 \): \[ \frac{3(0) + 7}{0 - 1} = \frac{7}{-1} = -7 < 2 \quad \text{(In solution)} \] 3. **Interval \((1, ∞)\):** Choose \( x = 2 \): \[ \frac{3(2) + 7}{2 - 1} = \frac{6 + 7}{1} = 13 > 2 \quad \text{(Not in solution)} \] ### Step 3: Write the Solution in Interval Notation The solution consists of all \( x \) values between \(-9\) and \(1\), excluding \(1\) where the expression is undefined. \[ \boxed{\,(-9,\ 1)\,} \] ### Graph of the Solution Set Below is a number line representation of the solution set: ``` <----|---------|---------|------> -9 1 (---------) ``` - **Open circles** at \( x = -9 \) and \( x = 1 \) indicate that these points are not included in the solution. - The **shaded region** between \(-9\) and \(1\) represents all values of \( x \) that satisfy the inequality.