Chapter 1

Determine whether each statement is true or false. If it is false, explain why. The union of the solution sets of \(x+1=6, x+1<6,\) and \(x+1>6\) is \((-\infty, \infty)\)

Complete each statement. The following key terms may be used once, more than once, or not at all. $$\begin{array}{cc}\text{linear equation}&\text{solution}&\text{algebraic expression}&\text{contradiction}& \text{all real numbers}\\\ \text{solution set}&\text{identity}&\text{conditional equation}&\text{first-degree equation }&\text{empty set}\varnothing \end{array}$$ A collection of numbers, variables, operation symbols, and grouping symbols, such as \(2(8 x-15),\) is a \((n)\) __________. While an equation (does/does not) include an equality symbol, there (is / is not) an equality symbol in an algebraic expression.

In each of the following, translate part (a) as an expression and translate part (b) as an equation or inequality. Use \(x\) to represent the number. (a) 15 more than a number (b) 15 is more than a number.

The distance between Cape Town, South Africa, and Miami is \(7700 \mathrm{mi}\). If a jet averages 550 mph between the two cities, what is its travel time in hours?

In each of the following, translate part (a) as an expression and translate part (b) as an equation or inequality. Use \(x\) to represent the number. (a) 5 greater than a number (b) 5 is greater than a number.

How many solutions will \(|a x+b|=k\) have for each situation? (a) \(k=0\) (b) \(k>0\) (c) \(k<0\)

Work each problem to review converting between decimals and percents. Write each decimal as a percent. (a) 0.35 (b) 0.18 (c) 0.02 (d) 0.075 (e) 1.5

Complete each statement. The following key terms may be used once, more than once, or not at all. $$\begin{array}{cc}\text{linear equation}&\text{solution}&\text{algebraic expression}&\text{contradiction}& \text{all real numbers}\\\ \text{solution set}&\text{identity}&\text{conditional equation}&\text{first-degree equation }&\text{empty set}\varnothing \end{array}$$ If we let \(x=2\) in the linear equation \(2 x+5=9,\) a (true/false) statement results. The number 2 is a(n) __________ of the equation, and \\{2\\} is the ____________.

Tri traveled from Chicago to Des Moines, a distance of \(300 \mathrm{mi}\), in \(10 \mathrm{hr}\). What was his rate in miles per hour?

In each of the following, translate part (a) as an expression and translate part (b) as an equation or inequality. Use \(x\) to represent the number. (a) 8 less than a number (b) 8 is less than a number.

Work each problem to review converting between decimals and percents. Write each percent as a decimal. (a) \(60 \%\) (b) \(37 \%\) (c) \(8 \%\) (d) \(3.5 \%\) (e) \(210 \%\)

Determine whether each statement is true or false. If it is false, explain why. The intersection of the sets \((-\infty, 7]\) and \([7, \infty)\) is \\{7\\} .

Complete each statement. The following key terms may be used once, more than once, or not at all. $$\begin{array}{cc}\text{linear equation}&\text{solution}&\text{algebraic expression}&\text{contradiction}& \text{all real numbers}\\\ \text{solution set}&\text{identity}&\text{conditional equation}&\text{first-degree equation }&\text{empty set}\varnothing \end{array}$$ A linear equation with one solution in its ____________, such as \(2 x+5=9,\) is \(\mathrm{a}(\mathrm{n})\) ___________.

In each of the following, translate part (a) as an expression and translate part (b) as an equation or inequality. Use \(x\) to represent the number. (a) 6 less than a number (b) 6 is less than a number.

A square has perimeter 160 in. What would be the perimeter of an equilateral triangle whose sides each measure the same length as the side of the square?

Solve each equation. $$ |x|=12 $$

Solve each formula for the specified variable. \(I=p r t\) for \(r \quad\) (simple interest)

Let \(A=\\{1,2,3,4,5,6\\}, B=\\{1,3,5\\}, C=\\{1,6\\},\) and \(D=\\{4\\} .\) Find each set. $$ A \cap D $$

An equilateral triangle has perimeter 27 in. What would be the area of a square whose sides each measure the same length as the side of the equilateral triangle?

Solve each equation. $$ |x|=14 $$

Solve each formula for the specified variable. \(d=r t\) for \(t \quad(\) distance \()\)

Let \(A=\\{1,2,3,4,5,6\\}, B=\\{1,3,5\\}, C=\\{1,6\\},\) and \(D=\\{4\\} .\) Find each set. $$ B \cap C $$

A student translated the phrase "the quotient of a number \(n\) and \(12^{\prime \prime}\) incorrectly as \(\frac{12}{n}\).Give the correct mathematical expression.

A circle has area \(25 \pi \mathrm{ft}^{2}\). What would be the perimeter of a square whose sides each measure the same length as the radius of the circle?

Solve each equation. $$ |4 x|=20 $$

Solve each formula for the specified variable. \(\mathscr{A}=L W \quad\) (area of a rectangle) (a) for \(W\) (b) for \(L\)

Let \(A=\\{1,2,3,4,5,6\\}, B=\\{1,3,5\\}, C=\\{1,6\\},\) and \(D=\\{4\\} .\) Find each set. $$ B \cap A $$

Which equations are linear equations in \(x ?\) A. \(3 x+x-2=0\) B. \(12=x^{2}\) C. \(9 x-4=9\) D. \(3 x+2 y=6\)

Translate each verbal phrase into \(a\) mathematical expression using \(x\) as the variable. $$ \text { Twice a number, decreased by } 13 $$

Solve each equation. $$ |5 x|=30 $$

Solve each formula for the specified variable. \(\mathscr{A}=b h \quad\) (area of a parallelogram) (a) for \(b\) (b) for \(h\)

Let \(A=\\{1,2,3,4,5,6\\}, B=\\{1,3,5\\}, C=\\{1,6\\},\) and \(D=\\{4\\} .\) Find each set. $$ C \cap A $$

Translate each verbal phrase into \(a\) mathematical expression using \(x\) as the variable. $$ \text { Triple a number, decreased by } 14 $$

Solve each equation. $$ |x-3|=9 $$

Solve each formula for the specified variable. \(P=2 L+2 W\) for \(L\) (perimeter of a rectangle)

Determine whether each of the following is an expression or an equation. \(-3 x+2-4=x\)

Let \(A=\\{1,2,3,4,5,6\\}, B=\\{1,3,5\\}, C=\\{1,6\\},\) and \(D=\\{4\\} .\) Find each set. $$ B \cap \varnothing $$

Translate each verbal phrase into \(a\) mathematical expression using \(x\) as the variable. $$ 12 \text { increased by four times a number } $$

Otis has a box of coins that he uses when he plays poker with his friends. The box contains 44 coins, consisting of pennies, dimes, and quarters. The number of pennies is equal to the number of dimes. The total value is \(\$ 4.37 .\) How many of each denomination of coin does he have in the box?

Solve each equation. $$ |x-5|=13 $$

Solve each formula for the specified variable. \(P=a+b+c \quad\) (perimeter of a triangle) (a) for \(b\) (b) for \(c\)

\(\quad\) A student solved the following inequality incorrectly as shown. $$ \begin{array}{l} -2 x<-18 \\ \frac{-2 x}{-2}<\frac{-18}{-2} \end{array} $$ $$ x<9 \quad(-\infty, 9) $$ WHAT WENT WRONG? Give the correct solution set.

Determine whether each of the following is an expression or an equation. \(-3 x+2-4-x=4\)

Let \(A=\\{1,2,3,4,5,6\\}, B=\\{1,3,5\\}, C=\\{1,6\\},\) and \(D=\\{4\\} .\) Find each set. $$ A \cap \varnothing $$

Translate each verbal phrase into \(a\) mathematical expression using \(x\) as the variable. $$ 15 \text { more than one-half of a number } $$

Nana found some coins under her sofa pillows. There were equal numbers of nickels and quarters and twice as many half-dollars as quarters. If she found \(\$ 2.60\) in all, how many of each denomination of coin did she find?

Solve each equation. $$ |2 x-1|=11 $$

Solve each formula for the specified variable. \(V=L W H\) (volume of a rectangular solid) (a) for \(W\) (b) for \(H\)

Let \(A=\\{1,2,3,4,5,6\\}, B=\\{1,3,5\\}, C=\\{1,6\\},\) and \(D=\\{4\\} .\) Find each set. $$ A \cup B $$

Determine whether each of the following is an expression or an equation. \(4(x+3)-2(x+1)-10\)