Question

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

$(x-4){(x+2)}^2<0$

Short answer

The solution set is$(-\infty ,-2)\cup (-2,4)$

Step-by-step solution

Step 1. Finding the interval.

Given inequality is$(x-4){(x+2)}^2<0$

factors on the left-hand side are $x-4,{(x+2)}^2$. These factors are zero

When $x=4,-2$respectively.

So, the numbers$4,-2$ divide the real line into three intervals$(-\infty ,-2),(-2,4),(4,\infty )$

Step 2. 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 the inequality is$(-\infty ,-2)\cup (-2,4)$

Step 4. Plotting the graph.