Exam 24: Inequalities and Linear Programming

Rewrite each pair of inequalities as a double inequality: (a) $x \geq 2$ and $x<15$ (b) $x>-4$ and $x \leq 25$ (c) $x \geq-5$ and $x \leq 12$ (d) $x>20$ and $x<64$

Rewrite each double inequality as two separate inequalities:

Graph the region in which both of the following two inequalities are satisfied: $y \geq x-8$ and $y \leq x / 2-6$

Write $(-3,8)$ as an inequality.

Graph the inequality on the number line: $-2 \leq x<1$

Maximize the function $z=4 x-3 y$ under the constraints $2 x+y \leq 8, x-2 y \leq 5,4 x-y\geq-16, x\geq 0, y\geq 0$ .

A company makes two different laptops. Model A requires 4 hours to make and $\$ 350$ in materials and yields a profit of $\$ 750$ . Model B requires 6 hours to make and $\$ 400$ in materials and yields a profit of $\$ 900$ . There are 1000 hours and $\$ 70000$ available per week to make laptops. How many of each laptop model must be made each month in order to maximize profit?

Solve the inequality: $2(4-x)+x \leq 5-2(1-2 x)$

Solve the inequality:

Solve the inequality: $4 x-8 \geq 3(4 x+1)+13$

Solve the inequality: $2 x^{2}+x \leq 15$

Maximize the function $z=-2 x+4 y$ under the constraints $x+y\leq 2,8 x-y\geq-8$ , and $y\geq 0$ .

Graph the inequality: $y\geq 2 x-1$

It costs a company $\$ 500$ in labor to make 75 parts, $\$ 200$ in material to make 30 parts, and there is an overhead cost of $\$ 24500$ a month. If each part is sold for $\$ 45$ , find the minimum number of parts that must be sold each month for the company to make a profit.

Solve the inequality: $|5 x-3|<2 x$

Maximize the function $z=3 x+2 y$ under the constraints $y-x \leq 1, x+2 y \leq 5, x\geq 0$ , and $y \geq 0$ .

Write using bracket notation.

Solve the inequality: $\frac{x^{2}-3 x}{2}>\frac{-x^{2}+5 x+90}{2}$

Solve the inequality: $3 x-9<4 x+2$