Section 1
Real Numbers and the Real Line
Express the given rational number as a repeating decimal. Use a bar to indicate the repeating digits.$$\frac{2}{9}$$
Express the given rational number as a repeating decimal. Use a bar to indicate the repeating digits.$$\frac{1}{11}$$
Express the given repeating decimal as a quotient of integers in lowest terms.$$0 . \overline{12}$$
Express the given repeating decimal as a quotient of integers in lowest terms.$$3.2 \overline{7}$$
Express the rational numbers $1 / 7,2 / 7,3 / 7,$ and $4 / 7$ as repeating decimals. (Use a calculator to give as many decimal digits as possible.) Do you see a pattern? Guess the decimal expansions of $5 / 7$ and $6 / 7$ and check your guesses.
Can two different decimals represent the same number? What number is represented by $0.999 \ldots=0 . \overline{9} ?$
Express the set of all real numbers $x$ satisfying the given conditions as an interval or a union of intervals.$$x \geq 0 \quad \text { and } \quad x \leq 5$$
Express the set of all real numbers $x$ satisfying the given conditions as an interval or a union of intervals.$$x < 2 \text { and } x \geq-3$$
Express the set of all real numbers $x$ satisfying the given conditions as an interval or a union of intervals.$$x > -5 \text { or } x < -6$$
Express the set of all real numbers $x$ satisfying the given conditions as an interval or a union of intervals.$$x \leq-1$$
Express the set of all real numbers $x$ satisfying the given conditions as an interval or a union of intervals.$$x > -2$$
Express the set of all real numbers $x$ satisfying the given conditions as an interval or a union of intervals.$$x < 4 \text { or } x \geq 2$$
Solve the given inequality, giving the solution set as an interval or union of intervals.$$-2 x>4$$
Solve the given inequality, giving the solution set as an interval or union of intervals.$$3 x+5 \leq 8$$
Solve the given inequality, giving the solution set as an interval or union of intervals.$$5 x-3 \leq 7-3 x$$
Solve the given inequality, giving the solution set as an interval or union of intervals.$$\frac{6-x}{4} \geq \frac{3 x-4}{2}$$
Solve the given inequality, giving the solution set as an interval or union of intervals.$$3(2-x) < 2(3+x)$$
Solve the given inequality, giving the solution set as an interval or union of intervals.$$x^{2} < 9$$
Solve the given inequality, giving the solution set as an interval or union of intervals.$$\frac{1}{2-x} < 3$$
Solve the given inequality, giving the solution set as an interval or union of intervals.$$\frac{x+1}{x} \geq 2$$
Solve the given inequality, giving the solution set as an interval or union of intervals.$$x^{2}-2 x \leq 0$$
Solve the given inequality, giving the solution set as an interval or union of intervals.$$6 x^{2}-5 x \leq-1$$
Solve the given inequality, giving the solution set as an interval or union of intervals.$$x^{3} > 4 x$$
Solve the given inequality, giving the solution set as an interval or union of intervals.$$x^{2}-x \leq 2$$
Solve the given inequality, giving the solution set as an interval or union of intervals.$$\frac{x}{2} \geq 1+\frac{4}{x}$$
Solve the given inequality, giving the solution set as an interval or union of intervals.$$\frac{3}{x-1} < \frac{2}{x+1}$$
Solve the equations.$$|x|=3$$
Solve the equations.$$|x-3|=7$$
Solve the equations.$$|2 t+5|=4$$
Solve the equations.$$|1-t|=1$$
Solve the equations.$$|8-3 s|=9$$
Solve the equations.$$\left|\frac{s}{2}-1\right|=1$$
Write the interval defined by the given inequality.$$|x| < 2$$
Write the interval defined by the given inequality.$$|x| \leq 2$$
Write the interval defined by the given inequality.$$|s-1| \leq 2$$
Write the interval defined by the given inequality.$$|t+2| < 1$$
Write the interval defined by the given inequality.$$|3 x-7| < 2$$
Write the interval defined by the given inequality.$$|2 x+5| < 1$$
Write the interval defined by the given inequality.$$\left|\frac{x}{2}-1\right| \leq 1$$
Write the interval defined by the given inequality.$$\left|2-\frac{x}{2}\right| < \frac{1}{2}$$
Solve the given inequality by interpreting it as a statement about distances on the real line.$$|x+1| > |x-3|$$
Solve the given inequality by interpreting it as a statement about distances on the real line.$$|x-3| < 2|x|$$
Do not fall into the trap $|-a|=a$. For what real numbers $a$ is this equation true? For what numbers is it false?
Solve the equation $|x-1|=1-x$
Show that the inequality$$|a-b| \geq|| a|-| b||$$holds for all real numbers $a$ and $b$