Chapter 1: Fundamentals

Simplify the fractional expression.

$\frac{{\left(7-3x\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}+\frac{3}{2}x{\left(7-3x\right)}^{-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{7-3x}$

Conjugate 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.

$z=\overline{z}\text{ifandonlyif}z\text{isreal.}$

Write each number in scientific notation.

1. $129,540,000$
2. $7,259,000,000$
3. $0.0000000014$
4. $0.0007029$

Let a, b, and c be real numbers such that$\mathbf{a}\mathbf{>}\mathbf{0}$, $\mathbf{b}\mathbf{<}\mathbf{0}$and$\mathbf{c}\mathbf{<}\mathbf{0}$. Find the sign of each expression.

$\mathbf{\left(}\mathbf{a}\mathbf{\right)}\mathbf{-}\mathbf{b}$

$\mathbf{\left(}\mathbf{b}\mathbf{\right)}\mathbf{a}\mathbf{+}\mathbf{bc}$

$\mathbf{\left(}\mathbf{c}\mathbf{\right)}\mathbf{c}\mathbf{-}\mathbf{a}$

$\mathbf{\left(}\mathbf{d}\mathbf{\right)}\mathbf{a}{\mathbf{b}}^2$

Find the number of real solutions of the equation $x^2+2.21x+1.21=0$by using discriminant.

Find the center and radius of the circle, and sketch its graph.

$x^2+y^2=5$

Geometry The hypotenuse of a right triangle has length $20\text{cm}$. The sum of the lengths of the other two sides is $28\text{cm}$. Find the lengths of the other two sides of the triangle.

Area of a Triangle Find the area of the triangle formed by the coordinate axes and the line

$2y+3x-6=0$

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

$\left|8x+3\right|>12$

Find the center and radius of the circle, and sketch its graph.

$x^2+{\left(y-4\right)}^2=1$