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

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

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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. The solution to the inequality $$ \frac{3x + 7}{x - 1} < 2 $$ is $$ x $$ in the interval $$ (-9, 1) $$.

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.

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