\( \frac{3 y+3}{4} \frac{-y-6}{3}>0 \) Show all your work And draw a number line First find LCO There is no.

To solve the inequality \[ \frac{3y + 3}{4} \cdot \frac{-y - 6}{3} > 0, \] we will follow these steps: 1. **Factor the expressions**. 2. **Identify critical points**. 3. **Test intervals**. 4. **Draw a number line**. ### Step 1: Factor the expressions First, we can factor the numerators: - The first term \(3y + 3\) can be factored as \(3(y + 1)\). - The second term \(-y - 6\) can be factored as \(-1(y + 6)\). Thus, we can rewrite the inequality as: \[ \frac{3(y + 1)}{4} \cdot \frac{-1(y + 6)}{3} > 0. \] ### Step 2: Simplify the expression Now, we can simplify the expression: \[ \frac{3(y + 1) \cdot -1(y + 6)}{4 \cdot 3} > 0. \] This simplifies to: \[ \frac{-(y + 1)(y + 6)}{4} > 0. \] ### Step 3: Identify critical points The critical points occur when the numerator is zero or undefined. The numerator is zero when: \[ -(y + 1)(y + 6) = 0. \] This gives us the critical points: 1. \(y + 1 = 0 \Rightarrow y = -1\) 2. \(y + 6 = 0 \Rightarrow y = -6\) The denominator \(4\) is never zero, so we only consider the critical points from the numerator. ### Step 4: Test intervals The critical points divide the number line into intervals: 1. \( (-\infty, -6) \) 2. \( (-6, -1) \) 3. \( (-1, \infty) \) We will test a point from each interval to determine where the inequality holds. - **Interval 1: \(y = -7\)** \[ \frac{-(-7 + 1)(-7 + 6)}{4} = \frac{-(-6)(-1)}{4} = \frac{-6}{4} < 0 \quad \text{(not valid)} \] - **Interval 2: \(y = -5\)** \[ \frac{-(-5 + 1)(-5 + 6)}{4} = \frac{-(-4)(1)}{4} = \frac{4}{4} > 0 \quad \text{(valid)} \] - **Interval 3: \(y = 0\)** \[ \frac{-(0 + 1)(0 + 6)}{4} = \frac{-(1)(6)}{4} = \frac{-6}{4} < 0 \quad \text{(not valid)} \] ### Step 5: Conclusion The inequality holds for the interval \( (-6, -1) \). ### Number Line Now, we can represent this on a number line: ``` <----|----|----|----|----|----|----|----|----|----> -7 -6 -5 -4 -3 -2 -1 0 1 (----------------) ``` The parentheses indicate that the endpoints \(-6\) and \(-1\) are not included in the solution. ### Final Answer The solution to the inequality is: \[ y \in (-6, -1). \]