Solve the inequality (1.5 + 0.1x)(800 - 20x) - 0.9(800 - 20x) is greater than or equal to 720.

Step-by-step solution

1. 1

   Spot the repeated bracket and factor it out. Both terms on the left contain $(800-20x)$, so pull it out instead of expanding twice:

   $\left(800-20x\right)\left[\left(1.5+0.1x\right)-0.9\right]=\left(800-20x\right)\left(0.6+0.1x\right).$

   In the original setting, $(1.5+0.1x)-0.9$ is the profit per unit and $(800-20x)$ is the number of units sold.

2. 2

   Expand the product.

   $\left(0.6+0.1x\right)\left(800-20x\right)=480-12x+80x-2x^{2}=-2x^{2}+68x+480.$

3. 3

   Move everything to one side.

   $-2x^{2}+68x+480\ge 720\ \Longrightarrow\ -2x^{2}+68x-240\ge 0.$

4. 4

   Divide by -2 and flip the inequality sign. Dividing an inequality by a negative number reverses it — the step most often forgotten here:

   $x^{2}-34x+120\le 0.$

5. 5

   Find the roots and read the interval.

   $x=\frac{34\pm\sqrt{1156-480}}{2}=\frac{34\pm\sqrt{676}}{2}=\frac{34\pm 26}{2}=4\ \text{or}\ 30.$

   An upward parabola is $\le 0$ between its roots, so

   $4\le x\le 30.$

6. 6

   Spot-check the endpoints and an interior point. At $x=4$: $(1.9)(720)-0.9(720)=(1.0)(720)=720$, exactly the threshold ✓. At $x=17$ (the vertex): $(3.2)(460)-0.9(460)=(2.3)(460)=1058\ge 720$ ✓. At $x=31$: $(4.6)(180)-0.9(180)=(3.7)(180)=666<720$, correctly outside.