The numerator of a fraction is five less than the denominator. The sum of the fraction and six times its reciprocal is $\frac { 25 } { 2 } .$ Find the numerator and denominator assuming that each is an integer.

Suppose a state’s income tax code states that tax liability is 12% on the first $20,000 of taxable earnings and 16% on the remainder. Find constants a and b for the tax function

$$T(x)=\left\{\begin{array}{ll}{0} & {\text { if } x \leq 0} \\ {a+0.12 x} & {\text { if } 0<x \leq 20,000} \\ {b+0.16(x-20,000)} & {\text { if } x>20,000}\end{array}\right.$$

such that T (x) is continuous for all x.

Suppose a state’s income tax code states that tax liability is 12% on the first $20,000 of taxable earnings and 16% on the remainder. Find constants a and b for the tax function

$$T(x)=\left\{\begin{array}{ll}{a+0.12 x} & {\text { if } x \leq 20,000} \\ {b+0.16(x-20,000)} & {\text { if } x>20,000}\end{array}\right.$$

such that $\lim _{x \rightarrow 0^{+}} T(x)=0$ and $\lim _{x \rightarrow 20,000} T(x)$ exists. Why is it important for these limits to exist?

Each of the following isotopes is used in medicine. Indicate the number of protons and neutrons in each isotope: chromium-51.

Use the test-point method to solve each polynomial inequality. $x ^ { 2 } + 4 x - 12 \leq 0$

To make a rectangular concrete patio. a homeowner used 70 feet of perimeter. into which she poured 138 cubic feet of concrete to form a slab 6 inches thick. What were the dimensions of the patio?

An air-conditioning repair bill for $229.50 showed a charge of$72.00 dollars for parts, with the remainder of the charge for labor. Let x = number of hours of labor it took to repair the air conditioner. Write an algebraic expression in x for the labor charge per hour.

Solve each inequality. Write the solution in interval notation. $\frac { 3 - 2 x } { 4 } + 1 > \frac { x - 5 } { 3 }$

Kevin won $735,000 on a TV game show. He decided to keep a certain amount for himself, give one-half the amount he kept for himself to his daughter, and give one-fourth the amount he kept for himself to his dad. How much did each person get?

The numerator of a fraction is two less than the denominator. The sum of the fraction and its reciprocal is $\frac { 25 } { 12 } .$ Find the numerator and denominator assuming that each is a positive integer.

The Beckly Company manufactures shaving sets for $3 each and sells them for$5 each. How many shaving sets must be sold for the company to recover an initial investment of $40,000 and earn an additional$30,000 as profit?

The sum of two numbers is 57, and their product is 782. Find the numbers.

Let z = a + bi and w = c + di. Prove each statement. $\operatorname { Im } ( z ) = \frac { z - \overline { z } } { 2 i }$

Find a and b. If - 4 < x < 0, then a < 3x + 4 < b.