Question

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

$(x+2)(x-1)(x-3)\le 0$

Short answer

The solution set is $(-\infty ,-2]\cup [1,3]$and the graph is

Step-by-step solution

Step 1. Finding the interval.

Given inequality is $(x+2)(x-1)(x-3)\le 0$

The factors on the left-hand side are $(x+2)$, $(x-1)$, and $(x-3)$. These factors are zero

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

So, the numbers $-2,1$and 3 divide the real number into 4 intervals

$(-\infty ,-2),(-2,1),(1,3)(3,\infty )$

Step 2. Determining the sign of each other.

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$x\le -2or1\le x\le 3$

i.e., $(-\infty ,-2]\cup [1,3]$

Step 4. Plotting the graph.