Chapter 1

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. Because there are four quarters in a dollar, I can use the formula \(q=4 d\) to determine the number of quarters, \(q,\) in \(d\) dollars.

Determine whether the given number is a solution of the equation. $$(y \div 6)+\frac{1}{3}=(y \div 2)-\frac{5}{9} ; 2 \frac{2}{3}$$

Describe what it means to raise a number to a power. In your description, include a discussion of the difference between \(-5^{2}\) and \((-5)^{2}\)

From here on, each exercise set will contain three review exercises. It is essential to review previously covered topics to improve your understanding of the topics and to help maintain your mastery of the material. If you are not certain how to solve a review exercise, turn to the section and the worked example given in parentheses at the end of each exercise. Determine whether this inequality is true or false: \(19 \geq-18 .\) (Section 1.3, Example 7)

What is a rational number?

Translate from English to an algebraic expression or equation, whichever is appropriate. Let the variable \(x\) represent the number. \(\frac{1}{5}\) of a number

Explain how to simplify \(4 x^{2}+6 x^{2} .\) Why is the sum not equal to \(10 x^{4} ?\)

From here on, each exercise set will contain three review exercises. It is essential to review previously covered topics to improve your understanding of the topics and to help maintain your mastery of the material. If you are not certain how to solve a review exercise, turn to the section and the worked example given in parentheses at the end of each exercise. Determine whether 18 is a solution of \(16=2(x-1)-x\) (Section 1.1, Example 4)

Explain how to express \(\frac{3}{8}\) as a decimal.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Some algebraic expressions contain the equality symbol, \(=\)

Translate from English to an algebraic expression or equation, whichever is appropriate. Let the variable \(x\) represent the number. \(\frac{1}{6}\) of a number

Why is the order of operations agreement needed?

Will help you prepare for the material covered in the next section. In each exercise, a subtraction is expressed as addition of an opposite. Find this sum, indicated by a question mark. $$7-10=7+(-10)=?$$

Describe the difference between a rational number and an irrational number.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The algebraic expressions \(3+2 x\) and \((3+2) x\) do not mean the same thing.

Translate from English to an algebraic expression or equation, whichever is appropriate. Let the variable \(x\) represent the number. A number decreased by \(\frac{1}{4}\) of itself

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. Without parentheses, an exponent has only the number next to it as its base.

Will help you prepare for the material covered in the next section. In each exercise, a subtraction is expressed as addition of an opposite. Find this sum, indicated by a question mark. $$-8-13=-8+(-13)=?$$

If you are given two different real numbers, explain how to determine which one is the lesser.

Translate from English to an algebraic expression or equation, whichever is appropriate. Let the variable \(x\) represent the number. A number decreased by \(\frac{1}{3}\) of itself

Will help you prepare for the material covered in the next section. In each exercise, a subtraction is expressed as addition of an opposite. Find this sum, indicated by a question mark. $$-8-(-13)=-8+13=?$$

In Exercises \(97-108,\) determine whether the given number is a solution of the equation. $$\frac{5 m-1}{6}=\frac{3 m-2}{4},-4$$

Describe what is meant by the absolute value of a number. Give an example with your explanation.

Translate from English to an algebraic expression or equation, whichever is appropriate. Let the variable \(x\) represent the number. A number decreased by \(\frac{1}{4}\) is half of that number.

I read that a certain star is \(10^{4}\) light-years from Earth, which means \(100,000\) light-years. When I evaluated \((-1)^{n},\) I obtained positive numbers when \(n\) was even and negative numbers when \(n\) was odd

Translate from English to an algebraic expression or equation, whichever is appropriate. Let the variable \(x\) represent the number. A number decreased by \(\frac{1}{3}\) is half of that number.

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. The rules for the order of operations avoid the confusion of obtaining different results when I simplify the same expression.

Will help you prepare for the material covered in the next section. In each exercise, use the given formula to perform the indicated operation with the two fractions. $$\frac{a}{b} \cdot \frac{c}{d}=\frac{a \cdot c}{b \cdot d} ; \quad \frac{3}{7} \cdot \frac{2}{5}$$

In Exercises \(109-116\), write a numerical expression for each phrase. Then simplify the numerical expression by performing the given operations. 8 added to the product of 4 and \(-10\)

Translate from English to an algebraic expression or equation, whichever is appropriate. Let the variable \(x\) represent the number. The sum of \(\frac{1}{7}\) of a number and \(\frac{1}{8}\) of that number gives 12.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. If \(x\) is \(-3,\) then the value of \(-3 x-9\) is \(-18\)

Will help you prepare for the material covered in the next section. In each exercise, use the given formula to perform the indicated operation with the two fractions. $$\frac{a}{b} \div \frac{c}{d}=\frac{a}{b} \cdot \frac{d}{c}=\frac{a \cdot d}{b \cdot c} ; \quad \frac{2}{3} \div \frac{7}{5}$$

In Exercises \(109-116\), write a numerical expression for each phrase. Then simplify the numerical expression by performing the given operations. 14 added to the product of 3 and \(-15\)

Translate from English to an algebraic expression or equation, whichever is appropriate. Let the variable \(x\) represent the number. The sum of \(\frac{1}{9}\) of a number and \(\frac{1}{10}\) of that number gives 15.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The algebraic expression \(\frac{6 x+6}{x+1}\) cannot have the same value when two different replacements are made for \(x\) such as \(x=-3\) and \(x=2\)

Will help you prepare for the material covered in the next section. In each exercise, use the given formula to perform the indicated operation with the two fractions. $$\frac{a}{b}-\frac{c}{b}=\frac{a-c}{b} ; \frac{9}{17}-\frac{5}{17}$$

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I evaluated the formula \(d=\sqrt{1.5 h}\) for a value of \(h\) that resulted in a rational number for \(d\).

In Exercises \(109-116\), write a numerical expression for each phrase. Then simplify the numerical expression by performing the given operations. The product of \(-9\) and \(-3,\) decreased by \(-2\)

Explain how to subtract real numbers.

Translate from English to an algebraic expression or equation, whichever is appropriate. Let the variable \(x\) represent the number. The product of \(\frac{2}{3}\) and a number increased by 6

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The value of \(\frac{|3-7|-2^{3}}{(-2)(-3)}\) is the fraction that results when \(\frac{1}{3}\) is subtracted from \(-\frac{1}{3}\)

In Exercises \(109-116\), write a numerical expression for each phrase. Then simplify the numerical expression by performing the given operations. The product of \(-6\) and \(-4,\) decreased by \(-5\)

How is \(4-(-2)\) read?

Translate from English to an algebraic expression or equation, whichever is appropriate. Let the variable \(x\) represent the number. The product of \(\frac{3}{4}\) and a number increased by 9

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Some whole numbers are not integers.

In Exercises \(109-116\), write a numerical expression for each phrase. Then simplify the numerical expression by performing the given operations. The quotient of \(-18\) and the sum of \(-15\) and 12

Explain how to simplify a series of additions and subtractions. Provide an example with your explanation.

Translate from English to an algebraic expression or equation, whichever is appropriate. Let the variable \(x\) represent the number. The product of \(\frac{2}{3}\) and a number, increased by \(6,\) is 3 less than the number.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\text { Simplify: } \frac{1}{4}-6(2+8) \div\left(-\frac{1}{3}\right)\left(-\frac{1}{9}\right)$$

In Exercises \(109-116\), write a numerical expression for each phrase. Then simplify the numerical expression by performing the given operations. The quotient of \(-25\) and the sum of \(-21\) and 16