Chapter 1

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

Determine whether the given number is a solution of the equation. $$4(6-z)+7 z=0 ;-8$$

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 for " 3 less than a number" is the same as the algebraic expression for "a number decreased by 3

Determine whether this inequality is true or false: \(19 \geq-18 .\) (Section 1.3, Example 7)

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 \(10 x^{4} ?\)

Determine whether the given number is a solution of the equation. $$5(7-z)+12 z=0 ;-5$$

Determine whether 18 is a solution of \(16=2(x-1)-x\)

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

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

Why is the order of operations agreement needed?

Determine whether the given number is a solution of the equation. $$14-2 x=-4 x+7 ;-2 \frac{1}{2}$$

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.

A subtraction is expressed as addition of an opposite. Find this sum, indicated by a question mark. 7-10=7+(-10)=?

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

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

Determine whether the given number is a solution of the equation. $$16-4 x=-2 x+21 ;-3 \frac{1}{2}$$

A subtraction is expressed as addition of an opposite. Find this sum, indicated by a question mark. -8-13=-8+(-13)=?

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

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

Determine whether the given number is a solution of the equation. $$\frac{5 m-1}{6}=\frac{3 m-2}{4} ;-4$$

A subtraction is expressed as addition of an opposite. Find this sum, indicated by a question mark. -8-(-13)=-8+13=?

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.

In Exercises \(106-109\), determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. When I evaluated \((-1)^{n}\), I obtained positive numbers when \(n\) was even and negative numbers when \(n\) was odd

Determine whether the given number is a solution of the equation. $$\frac{6 m-5}{11}=\frac{3 m-2}{5} ;-1$$

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.

In Exercises \(106-109\), 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. $$\frac{a}{b} \cdot \frac{c}{d}=\frac{a \cdot c}{b \cdot d} ; \quad \frac{3}{7} \cdot \frac{2}{5}$$

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 .

In Exercises \(110-113\), 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. $$\frac{a}{b} \div \frac{c}{d}=\frac{a}{b} \cdot \frac{d}{c}=\frac{a \cdot d}{b \cdot c} ; \quad \frac{2}{3} \div \frac{7}{5}$$

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

Explain how to subtract real numbers.

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 \(d\)

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. $$\frac{a}{b}-\frac{c}{b}=\frac{a-c}{b} ; \frac{9}{17}-\frac{5}{17}$$

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 \(-2\)

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

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

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 \(x=2\) The value of \(\frac{|3-7|-2^{3}}{(-2)(-3)}\) is the fraction that results when \(\frac{1}{3}\) is subtracted from \(-\frac{1}{3}\)

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Every rational number is an integer.

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 \(-5\)

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

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

In Exercises \(110-113\), determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\begin{aligned} -2\left(6-4^{2}\right)^{3} &=-2(6-16)^{3} \\ &=-2(-10)^{3}=(-20)^{3}=-8000 \end{aligned}$$

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.

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

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.

Explain how to find the terms of the algebraic expression \(5 x-2 y-7\).