Question

Graph each inequality.

$3x+9y\ge -15$

Short answer

The graph of the inequality is provided below.

Step-by-step solution

Step 1. Apply the concept of graphing the inequality

The steps to graph the inequality are provided below.

1. If the inequality contains greater than or less than sign then the boundary of the line will be dashed. If the inequality contains signs of greater than or equal to or less than or equal to then the boundary of the line will be solid.

2. Select a point (known as test point) from the plane that does not lie on the boundary on the line and substitute it in the inequality.

3. If the inequality is true then shade the region that contains the test point otherwise shade the other region when inequality is false.

Step 2. Interpret the sign of the inequality

Consider the inequality provided below.

$3x+9y\ge -15$

The inequality contains the sign of greater than or equal to.

Therefore, the boundary line will be solid.

Step 3. Convert the inequality to slope intercept form

The corresponding equation is $3x+9y\ge -15$.

Equation of line in slope intercept form is expressed below.

$y=mx+c$

Where m is the slope and c is the intercept of y-axis.

Rewrite the equation as provided below.

$\begin{array}{c}y=\frac{-15-3x}{9}\\ y=-\frac{1}{3}x-\frac{5}{3}\end{array}$

Now, the equation is in the form $y=mx+c$. Here slope m of the line is $-\frac{1}{3}$ and intercept of y-axis c is $-\frac{5}{3}$.

Plot the equation.

Step 4. Shade the region

Take a test point that does not lie on the boundary of the line, say $\left(0,0\right)$.

Substitute the point $\left(0,0\right)$ in the inequality and check whether it’s true or not.

$\begin{array}{c}3x+9y\ge -15\\ 3\left(0\right)+9\left(0\right)\ge -15\\ 0+0\ge -15\\ 0\ge -15\end{array}$

This is true.

Therefore, shade the region containing the point $\left(0,0\right)$.

Hence, the graph of the inequality $3x+9y\ge -15$ is provided below.