Questions: Solve the inequality. 3 x^2 + 7 x + 2 ≤ 0 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution is the interval . (Type your answer in interval notation. Simplify your answer. Use integers or fractions for any numbers in the expression.) B. The solution is the list of x-values x= . (Simplify your answer. Type an integer or a fraction. Use a comma to separate answers as needed.) C. There is no solution.

$Solve the inequality. 3 x^2 + 7 x + 2 ≤ 0 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution is the interval . (Type your answer in interval notation. Simplify your answer. Use integers or fractions for any numbers in the expression.) B. The solution is the list of x-values x= . (Simplify your answer. Type an integer or a fraction. Use a comma to separate answers as needed.) C. There is no solution.$

Transcript text: Solve the inequality. \[ 3 x^{2}+7 x+2 \leq 0 \] Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution is the interval $\square$ . (Type your answer in interval notation. Simplify your answer. Use integers or fractions for any numbers in the expression.) B. The solution is the list of $x$-values $x=$ $\square$ . (Simplify your answer. Type an integer or a fraction. Use a comma to separate answers as needed.) C. There is no solution.

Solution

Solution Steps

Step 1: Identify the quadratic inequality

We are given the quadratic inequality: \[ 3x^2 + 7x + 2 \leq 0 \]

Step 2: Find the roots of the quadratic equation

To solve the inequality, we first need to find the roots of the corresponding quadratic equation: \[ 3x^2 + 7x + 2 = 0 \] We use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where $a = 3$, $b = 7$, and $c = 2$.

Step 3: Calculate the discriminant

First, we calculate the discriminant: \[ \Delta = b^2 - 4ac = 7^2 - 4 \cdot 3 \cdot 2 = 49 - 24 = 25 \]

Step 4: Find the roots

Since the discriminant is positive, we have two distinct real roots: \[ x = \frac{-7 \pm \sqrt{25}}{2 \cdot 3} = \frac{-7 \pm 5}{6} \] Thus, the roots are: \[ x_1 = \frac{-7 + 5}{6} = \frac{-2}{6} = -\frac{1}{3} \] \[ x_2 = \frac{-7 - 5}{6} = \frac{-12}{6} = -2 \]

Step 5: Determine the intervals to test

The roots divide the number line into three intervals:

$ (-\infty, -2) $
$ (-2, -\frac{1}{3}) $
$ (-\frac{1}{3}, \infty) $

Step 6: Test the intervals

We need to determine where the quadratic expression $3x^2 + 7x + 2$ is less than or equal to zero.

For $x \in (-\infty, -2)$, choose $x = -3$: \[ 3(-3)^2 + 7(-3) + 2 = 27 - 21 + 2 = 8 \quad (\text{positive}) \]
For $x \in (-2, -\frac{1}{3})$, choose $x = -1$: \[ 3(-1)^2 + 7(-1) + 2 = 3 - 7 + 2 = -2 \quad (\text{negative}) \]
For $x \in (-\frac{1}{3}, \infty)$, choose $x = 0$: \[ 3(0)^2 + 7(0) + 2 = 2 \quad (\text{positive}) \]

Step 7: Combine the intervals

The quadratic expression $3x^2 + 7x + 2$ is less than or equal to zero in the interval $(-2, -\frac{1}{3})$. Since the inequality is $\leq 0$, we include the roots $-2$ and $-\frac{1}{3}$.