Chapter 1

The troposphere is the layer of atmosphere closest to Earth. The average upper boundary of the layer is about 13 kilometers above Earth’s surface. This height varies with latitude and with the seasons by as much as 5 kilometers. Write and solve an equation describing the maximum and minimum heights of the upper bound of the troposphere.

Evaluate each expression. \(-\sqrt{25}\)

For Exercises \(46-49,\) use the following information. You can use the operators in the LOGIC menu on the TI-83/84 Plus to graph compound and absolute value inequalities. To display the LOGIC menu, press 2nd Test. Write the expression you would enter for \(Y 1\) to find the solution set of the compound inequality \(5 x+2 \geq 3\) or \(5 x+2 \leq-3\) . Then use the graphing calculator to find the solution set.

Define a variable and write an inequality for each problem. Then solve. 9 less than a number is at most that same number divided by 2.

NUMBER THEORY For Exercises \(46-49,\) use the properties of real numbers to answer each question. If \(m n=1,\) what is the value of \(n ?\) What is \(n^{\prime}\) s relationship to \(m ?\)

Write a verbal expression to represent each equation. $$ \frac{b}{4}=2(b+1) $$

Evaluate each expression. \(\sqrt{\frac{4}{9}}\)

For Exercises \(46-49,\) use the following information. You can use the operators in the LOGIC menu on the TI-83/84 Plus to graph compound and absolute value inequalities. To display the LOGIC menu, press 2nd Test. A graphing calculator can also be used to solve absolute value inequalities. Write the expression you would enter for \(Y 1\) to find the solution set of the inequality \(|2 x-6| > 10 .\) Then use the graphing calculator to find the solution set. (Hint: The absolute value operator is item 1 on the MATH NUM menu.)

NUMBER THEORY For Exercises \(46-49,\) use the properties of real numbers to answer each question. If \(m n=m\) and \(m \neq 0,\) what is the value of \(n ?\)

Write a verbal expression to represent each equation. $$ 7-\frac{1}{2} x=\frac{3}{x^{2}} $$

Determine whether each statement is sometimes, always, or never true. Explain your reasoning. If \(a\) and \(b\) are real numbers, then \(|a+b|=|a|+|b|\)

Evaluate each expression. \(\sqrt{\frac{36}{49}}\)

Write a compound inequality for which the graph is the empty set.

Flavio’s scores on the first four of five 100-point history tests were 85, 91, 89, and 94. If a grade of at least 90 is an A, write an inequality to find the score Flavio must receive on the fifth test to have an A test average.

MATH HISTORY For Exercises \(50-52\) , use the following information. The Greek mathematician Pythagoras believed that all things could be described by numbers. By number he meant a positive integer. To what set of numbers was Pythagoras referring when he spoke of numbers?

Solve each equation or formula for the specified variable. $$ \frac{a(b-2)}{c-3}=x, \text { for } b $$

Determine whether each statement is sometimes, always, or never true. Explain your reasoning. If \(a, b,\) and \(c\) are real numbers, then \(c|a+b|=|c a+c b|\)

MATH HISTORY For Exercises \(50-52\) , use the following information. The Greek mathematician Pythagoras believed that all things could be described by numbers. By number he meant a positive integer. Use the formula \(c=\sqrt{2 s^{2}}\) to calculate the length of the hypotenuse \(c,\) or longest side, of this right triangle using \(s,\) the length of one leg.

Solve each equation or formula for the specified variable. $$ x=\frac{y}{y+4}, \text { for } y $$

Determine whether each statement is sometimes, always, or never true. Explain your reasoning. For all real numbers \(a\) and \(b, a \neq 0,\) the equation \(|a x+b|=0\) will have exactly one solution.

CHALLENGE Graph each set on a number line. a. \(-2 < x < 4\) b. \(x < -1\) or \(x > 3\) c. \((-2 < x < 4)\) and \((x < -1 \text { or } x >3)\) (Hint: This is the intersection of the graphs in part a and part b. d. Solve \(3 < |x+2| \leq 8 .\) Explain your reasoning and graph the solution set.

Use a graphing calculator to solve each inequality. \(-5 x-8<7\)

Solve each equation. Check your solution. $$ \frac{1}{9}-\frac{2}{3} b=\frac{1}{18} $$

Use a graphing calculator to solve each inequality. \(-4(6 x-3) \leq 60\)

Name the sets of numbers to which each number belongs. $$ 0 $$

Solve each equation. Check your solution. $$ 3 f-2=4 f+5 $$

If \(5 < a < 7 < b < 14\) then which of the following best describes \(\frac{a}{b} ?\) A \(\frac{5}{7} < \frac{a}{b} < \frac{1}{2}\) B \(\frac{5}{14} < \frac{a}{b} < \frac{1}{2}\) C \(\frac{5}{7} < \frac{a}{b} < 1\) D \(\frac{5}{14} < \frac{a}{b} < 1\)

Use a graphing calculator to solve each inequality. \(3(x+3) \geq 2(x+4)\)

Solve each equation. Check your solution. $$ 4(k+3)+2=4.5(k+1) $$

For a party, Lenora bought several pounds of cashews and several pounds of almonds. The cashews cost \(\$ 8\) per pound, and the almonds cost \(\$ 6\) per pound. Lenora bought a total of 7 pounds and paid a total of \(\$ 48 .\) How many pounds of cashews did she buy? F. 2 pounds G. 3 pounds H. 4 pounds J. 5 pounds

What is the solution set of the inequality \(-20 < 4 x-8 < 12 ?\) \(\mathbf{F}-7 < x < 1\) \(\mathbf{G}-3 < x < 5\) \(\mathbf{H}-7 < x < 5\) \(\mathbf{J}-3 < x < 1\)

Solve each equation. Check your solution. $$ 4.3 n+1=7-1.7 n $$

Solve each equation. Check your solution. \(3 x+6=22\)

Solve each inequality. Then graph the solution set on a number line. (lesson \(1-5 )\) $$ 2 d+15 \geq 3 $$

Name the sets of numbers to which all of the following numbers belong. Then arrange the numbers in order from least to greatest. $$2.49,2.4 \overline{9}, 2.4,2.49,2 . \overline{9}$$

Solve each equation. Check your solution. $$ \frac{3}{11} a-1=\frac{7}{11} a+9 $$

Solve each equation. Check your solution. \(7 p-4=3(4+5 p)\)

Solve each inequality. Then graph the solution set on a number line. (lesson \(1-5 )\) $$ 7 x+11 > 9 x+3 $$

Which of the following properties hold for inequalities? Explain your reasoning or give a counterexample. a. Reflexive b. Symmetric c. Transitive

Give an example of a number that satisfies each condition. integer, but not a natural number

Solve each equation. Check your solution. $$ \frac{2}{5} x+\frac{3}{7}=1-\frac{4}{7} x $$

Solve each equation. Check your solution. \(\frac{5}{7} y-3=\frac{3}{7} y+1\)

Solve each inequality. Then graph the solution set on a number line. (lesson \(1-5 )\) $$ 3 n+4(n+3) < 5(n+2) $$

Name the property illustrated by each equation. \((5+9)+13=13+(5+9)\)

To get a chance to win a car, you must guess the number of keys in a jar to within 5 of the actual number. Those who are within this range are given a key to try in the ignition of the car. Suppose there are 587 keys in the jar. Write and solve an equation to determine the highest and lowest guesses that will give contestants a chance to win the car. (Lesson \(1-4 )\)

Determine whether each statement is true or false. If false, give a counterexample. A counterexample is a specific case that shows that a statement is false. Every whole number is an integer.

For Exercises \(58-63,\) define a variable, write an equation, and solve the problem. Schoot A school conference room can seat a maximum of 83 people. The principal and two counselors need to meet with the school's student athletes to discuss eligibility requirements. If each student must bring a parent with them, how many students can attend each meeting?

Name the property illustrated by each equation. \(m(4-3)=m \cdot 4-m \cdot 3\)

Solve each equation. Check your solutions. $$ 5|x-3|=65 $$

If \(ab c\) II. \(a+cb-c\) A. I only B. II only C. III only D. I and II only