Questions: Make a sign chart that represents the expression 7x^3/(3x+6)<0 and use your sign chart to solve the inequality. Plot x-values for which the expression equals zero or is undefined, then click next. Click to plot points. Click points to delete them.

Make a sign chart that represents the expression 7x^3/(3x+6)<0 and use your sign chart to solve the inequality.

Plot x-values for which the expression equals zero or is undefined, then click next.

Click to plot points. Click points to delete them.

Transcript text: Make a sign chart that represents the expression $\frac{7 x^{3}}{3 x+6}<0$ and use your sign chart to solve the inequality. Plot x-values for which the expression equals zero or is undefined, then click next. Click to plot points. Click points to delete them.

Solution

Solution Steps

Step 1: Find the values that make the numerator zero

The numerator of the expression is $7x^3$. To find the values that make the numerator zero, set $7x^3 = 0$: \[ 7x^3 = 0 \] \[ x = 0 \]

Step 2: Find the values that make the denominator zero

The denominator of the expression is $3x + 6$. To find the values that make the denominator zero, set $3x + 6 = 0$: \[ 3x + 6 = 0 \] \[ 3x = -6 \] \[ x = -2 \]

Step 3: Plot the critical points on the number line

The critical points are $x = 0$ and $x = -2$. These points divide the number line into three intervals: $(-\infty, -2)$, $(-2, 0)$, and $(0, \infty)$.