Questions: Solve the polynomial inequality and graph the solution set on a real number line. Express the solution set in interval notation. 3x^2+7x<6 Solve the inequality. What is the solution set? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. The solution is . A. (Simplify your answer. Type your answer in interval notation. Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression.) B. The solution set is the empty set.

$Solve the polynomial inequality and graph the solution set on a real number line. Express the solution set in interval notation. 3x^2+7x<6 Solve the inequality. What is the solution set? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. The solution is . A. (Simplify your answer. Type your answer in interval notation. Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression.) B. The solution set is the empty set.$

Transcript text: Solve the polynomial inequality and graph the solution set on a real number line. Express the solution set in interval notation. \[ 3 x^{2}+7 x<6 \] Solve the inequality. What is the solution set? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. The solution is $\square$ $\square$. A. (Simplify your answer. Type your answer in interval notation. Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression.) B. The solution set is the empty set.

Solution

Solution Steps

Step 1: Rewrite the Inequality in Standard Form

First, we need to rewrite the given inequality in standard form: \[ 3x^2 + 7x < 6 \] Subtract 6 from both sides to set the inequality to zero: \[ 3x^2 + 7x - 6 < 0 \]

Step 2: Solve the Corresponding Equation

Next, we solve the corresponding equation to find the critical points: \[ 3x^2 + 7x - 6 = 0 \] We use the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a = 3$, $b = 7$, and $c = -6$: \[ x = \frac{-7 \pm \sqrt{7^2 - 4 \cdot 3 \cdot (-6)}}{2 \cdot 3} \] \[ x = \frac{-7 \pm \sqrt{49 + 72}}{6} \] \[ x = \frac{-7 \pm \sqrt{121}}{6} \] \[ x = \frac{-7 \pm 11}{6} \] This gives us two solutions: \[ x = \frac{-7 + 11}{6} = \frac{4}{6} = \frac{2}{3} \] \[ x = \frac{-7 - 11}{6} = \frac{-18}{6} = -3 \]

Step 3: Determine the Intervals

The critical points divide the real number line into three intervals:

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

Step 4: Test the Intervals

We test a point from each interval to determine where the inequality $3x^2 + 7x - 6 < 0$ holds.

For $x = -4$ in the interval $(- \infty, -3)$: \[ 3(-4)^2 + 7(-4) - 6 = 48 - 28 - 6 = 34 \quad (\text{not less than 0}) \]
For $x = 0$ in the interval $(-3, \frac{2}{3})$: \[ 3(0)^2 + 7(0) - 6 = -6 \quad (\text{less than 0}) \]
For $x = 1$ in the interval $(\frac{2}{3}, \infty)$: \[ 3(1)^2 + 7(1) - 6 = 3 + 7 - 6 = 4 \quad (\text{not less than 0}) \]

Step 5: Write the Solution in Interval Notation

The inequality $3x^2 + 7x - 6 < 0$ holds in the interval $(-3, \frac{2}{3})$.