Chapter 1: Fundamentals

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

$\left|x-5\right|\le 3$

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.

${\left(\overline{z}\right)}^2=\overline{z^2}$

Yes or No? If No, give a reason.

(a) Is the expression${\left(x+5\right)}^2$equal to$x^2+25$?

(b) When you expand${\left(x+a\right)}^2$, where$a\ne 0$, do you get three terms?

(c) Is the expression$\left(x+5\right)\left(x-5\right)$ equal to$x^2-25$?

(d) When you expand$\left(x+a\right)\left(x-a\right)$, where$a\ne 0$, do you get two terms?

Noise and Intelligibility Audiologists study the intelligibility of spoken sentences under different noise levels. Intelligibility, the MRT score, is measured as the percent of a spoken sentence that the listener can decipher at a certain noise level in decibels (dB). The table shows the results of one such test.

(a) Make a scatter plot of the data.

(b) Find and graph the regression line.

(c) Find the correlation coefficient. Is a linear model appropriate?

(d) Use the linear model in part (b) to estimate the intelligibility of a sentence at a 94-dB noise level.

To eliminate the square root in the equation $2x+1=\sqrt{x+1}$, we ___ each side to get the equation ___

Express the inequality in interval notation, and then graph the corresponding interval.

$-1<x\le 5$

Real and Imaginary Parts Find the real and imaginary parts of the complex number.

$-6+4i$

(a) What does$\sqrt[n]{a}=b$ mean?

(b) Is it true that$\sqrt{a^2}$ is equal to $\left|a\right|$? Try values for a that are positive and negative.

(c) How many real$n^{th}$ root does a positive real number if n is even? If n is odd?

(c) What does$a^{-n}$ mean if n is negative integer? What does$3^{-2}$ mean?

(d) Is$\sqrt[4]{-2}$ a real number? Is$\sqrt[3]{-2}$ a real number? Explain why or why not.

Sketch a graph of lines $y=-3$and $x=-3$. Are the lines perpendicular?

Solve the equation both algebraically and graphically. (Lesson 1.11)

$\frac{4}{x+2}-\frac{6}{2x}=\frac{5}{2x+4}$