Question

Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set.

$4x^2(x^2-9)\le 0$

Short answer

The solution set is$[-3,3]$

Step-by-step solution

Step 1. Finding the interval.

Given inequality is $4x^2(x^2-9)\le 0$

The factors on the left-hand side are $x^2,(x+3),(x^2-9)$. These factors are zero

When $x=-2,1$and 3 respectively.

So, the numbers$0,-3,+3$ divide the real line into four intervals$(-\infty ,-3),(-3,0),(0,3),(3,\infty )$

Determining the sign of each factor.

To determine the sign of each factor on each of the intervals that we found, we use test values. We choose a number inside each intervals and check the sign of each factors.

Step 3. Making the table.

So, the required solution of or the inequality is$[-3,0]\cup [0,3]$

i.e,$[-3,3]$

Step 4. Plotting the graph.

Graph of the inequality is