91Ó°ÊÓ

Solve compound inequality. \(-11<2 x-1 \leq-5\)

The mathematician Girolamo Cardano is credited with the first use (in 1545 ) of negative square roots in solving the now-famous problem, "Find two numbers whose sum is 10 and whose product is \(40 .^{\prime \prime}\) Show that the complex numbers \(5+i \sqrt{15}\) and \(5-i \sqrt{15}\) satisfy the conditions of the problem. (Cardano did not use the symbolism \(i \sqrt{15}\) or even \(\sqrt{-15} .\) He wrote \(\mathrm{Rm} 15\) for \(\sqrt{-15},\) meaning "radix minus 15." He regarded the numbers \(5+\) R.m 15 and \(5-\) R.m 15 as "fictitious" or "ghost numbers," and considered the problem "manifestly impossible." But in a mathematically adventurous spirit, he exclaimed, "Nevertheless, we will operate."

A stand-up comedian uses algebra in some jokes, including one about a telephone recording that announces "You have just reached an imaginary number. Please multiply by \(i\) and dial again." Explain the joke.

Solve each absolute value equation or indicate that the equation has no solution. $$|2 x-1|+3-3$$

Exercises \(78-80\) will help you prepare for the material covered in the next section. Factor: $$ 2 x^{2}+7 x-4 $$

Solve absolute value inequality. \(5|2 x+1|-3 \geq 9\)

Solve absolute value inequality. \(5>|4-x|\)

Suppose that we agree to pay you 8ç for every problem in this chapter that you solve correctly and fine you 5 ç for every problem done incorrectly. If at the end of 26 problems we do not owe each other any money, how many problems did you solve correctly?

Solve absolute value inequality. \(12<\left|-2 x+\frac{6}{7}\right|+\frac{3}{7}\)

Solve each equation. $$x(x+1)^{3}-42(x+1)^{2}-0$$