Chapter 1: Fundamentals

Simplify the fractional expression.

$\sqrt{1+{\left(\frac{x}{\sqrt{1-x^2}}\right)}^2}$

Recall that the symbol $\overline{z}$represents the complex conjugate of z. If $z=a+bi\text{and}w=c+di$, show that each statement is true.

$\overline{z}+\overline{w}=\overline{z+w}$

Find the real solutions of the quadratic equation $w^2=3\left(w-1\right)$.

A plank 30 ft long rests on top of a flat roofed building, with 5 ft of the plank projecting over the edge, as shown in the figure. A worker weighing 240 lb sits on one end of the plank. What is the largest weight that can be hung on the projecting end of the plank if it is to remain in balance? (Use the law of lever stated in Exercise 77).

An equation and its graph are given. Find the x- and y-intercepts.

$x^2+y^3-x^2{y}^2=64$

Use a Special Factoring Formula to factor the expression

${\left(x+3\right)}^2-4$

Find the real and complex solution of the equation $2x^2-3x+2=0$.

Simplify the fractional expression.

$\sqrt{1+{\left(x^3-\frac{1}{4x^3}\right)}^2}$

Express each repeating decimal as a fraction.

$\mathbf{\left(}\mathbf{a}\mathbf{\right)}\mathbf{5}\mathbf{.}\overline{23}$

$\mathbf{\left(}\mathbf{b}\mathbf{\right)}\mathbf{1}.3\overline{7}$

$\mathbf{\left(}\mathbf{c}\mathbf{\right)}\mathbf{2}.1\overline{35}$

Solve the absolute value inequality. Express the answer using interval notation and graph the solution set.

$\frac{1}{2}\left|x\right|\ge 1$