Chapter 1

Tools of Algebra

In Chapter 1, You Will…

Review and extend your knowledge of algebraic expressions and your skills in solving equations and inequalities.

Solve absolute value equations and inequalities by changing them to compound equations and inequalities.

1-1 Properties of Real Numbers

What You’ll Learn …

To graph and order real numbersTo identify and use properties of real

numbers

Number Classification

• Natural numbers are the counting numbers. 1,2,3,4,…..• Whole numbers are natural numbers and zero. 0,1,2,3,4,….• Integers are whole numbers and their opposites. ….-3,-2,-

1,0,1,2,3,….• Rational numbers can be written as a fraction. 7/5, -3/2,

0, 0.3, -1.2, 9• Irrational numbers cannot be written as a fraction. 2 ,

7 , • All of these numbers are real numbers.

Real Numbers

Rational Numbers Irrational Numbers

Integers

Whole Numbers

04/2

625

-4

-10 5

-3

58

-2 3 0.37

0.31

10

- 123

23

0.1011001000…

Example 1: Real World Connection

Many mathematical relationships involving variables are related to amusement parks. Which set of numbers best describes the values for each variable?

a. The cost C of admission for n people

b. The maximum speed s in meters per second on a roller coaster of height h in meters

c. The park’s profit (or loss) P in dollars for each week w of the year

Example 2: Graphing Numbers on a Number Line

- 3 1.7 5

2

Change toDecimals

Example 3: Ordering Real Numbers

An inequality is a mathematical sentence that compares the value of two expressions using an inequality symbol.

< Less than

< Less than or equal to

> Greater than

> Greater than or equal to

Compare - 0.25 and - 0.01

The opposite or additive inverse of any number a is –a. The sum of opposites is 0.

Find the opposite of a. -3.2b. 400c. 1/5

The reciprocal or multiplicative inverse of any nonzero number a is 1/a. The product of reciprocals is 1.

Find reciprocal of a. -3.2b. 400c. 1/5

Example 4: Finding Inverses

Properties of Real Numbers

Property Addition Multiplication

Closure a +b is a real number ab is a real number

Commutative a + b = b + a ab = ba

Associative (a+b)+c = a+(b+c) (ab)c = a(bc)

Identity a+0=a, 0+a=a a(1)=a, 1(a)=a

Inverse a +(-a) = 1 a(1/a)= 1, a ≠0

a(b+c) = ab+acDistributive

Example 5: Identify Properties of Real Numbers

Which property is illustrated?

a. 6+(-6) = 0

b. (-4 1) - 2 = -4 – 2

c. (3+0) - 5 = 3 – 5

d. -5 + [2 +(-3)] = (-5 + 2) + (-3)

*

Example 6: Finding Absolute Value

The absolute value of a real number is its distance from zero on the number line.

Find the absolute value of:

a. -4

b. 0

c. -1 (-2)

1-2 Algebraic Expressions

1.03 Operate with algebraic expressions (polynomial, rational, complex fractions) to solve problems.

What you’ll learn … To evaluate algebraic

expressions To simplify algebraic

expressions

Evaluating Algebraic Expressions

A variable is a symbol, usually a letter, that represents one or more numbers.

An expression that contains one or more variables is an algebraic expression or a variable expression.

When you substitute numbers for the variables in an expression and follow the order of operations, you evaluate the expression.

Example 1- Evaluating an Algebraic Expression

For x = 4 and y = -2

x + y ÷ x

3x – 4y + x – y

x + 2x ÷ y - 2y

Example 2- Evaluating an Algebraic Expression with Exponents

For c = -3 and d = 5

c2 - d2

c(3 – d) – c2

-d2 - 4(d – 2c)

Example 3 Real World Connection

The expression -3y +61 models the percent of eligible voters who voted in presidential elections from 1960 to 2000. In the expression, y represents the number of years since 1960. Find the approximate percent of eligible voters who voted in 1988.

Simplifying Algebraic Expressions

In an algebraic expression such as -4x +10, the parts that are added are called terms. A term is a number, a variable, or the product of a number and one or more variables.

The numerical factor in a term is the coefficient.

Like terms have the same variables raised to the same powers.

Example 4 Combining Like Terms

3k - k

5z2 – 10z – 8z2 + z

-(m – n) +2(m – 3n)

y(1 + y)- 3y2 – (y + 1)

Example 5 Finding Perimeter

3x

3x

2x – y

y

5x – 2y

2x

3c

2d

3c

d

d

6c – 2d

6c – 2d

8c +d

1-3 Solving Equations

1.03 Operate with algebraic expressions (polynomial, rational, complex fractions) to solve problems.

What you’ll learn … To solve equations To solve problems by

writing equations

Solving Equations

An equations that contains a variable may be true for some replacements of the variable and false for others.

A number that makes the equation true is a solution of the equation.

Properties of Equality

Reflexive Property a = a Symmetric Property If a=b then b=a Transitive Property If a=b and b=c then a=c Addition Property If a=b then a+c = b+c Subtraction Property If a=b then a-c = b-c Multiplication Property If a=b then ac = bc Division Property If a=b and c≠0 then a/c = b/c Substitution Property If a=b then b can be substituted for a in any

expression to obtain an equivalent expression.

Steps to Solving Equations

● Simplify each side of the equation, if needed, by distributing or combining like terms.

● Move variables to one side of the equation by using the opposite operation of addition or subtraction.

● Isolate the variable by applying the opposite operation to each side.• First, use the opposite operation of addition or

subtraction. • Second, use the opposite operation of

multiplication or division.● Check your answer.

Example 1 Solving an Equation with a Variable on Both Sides

13y + 48 = 8y - 47 2t – 3 = 9 – 4t

Example 2 Using the Distributive Property

3x – 7(2x – 13) = 3(-2x +9)

6(t – 2) = 2(9 – 2t)

Example 3 Solving a Formula for One of Its Variables

Solve for h.

A = ½ h (b1 + b2)

Solve for x.

ax +bx – 15 = 0

Example 4 Solving an Equation for One of Its Variables

Example 5 Real World Connection

A dog kennel owner has 100 ft of fencing to enclose a rectangular dog run. She wants it to be 5 times as long as it is wide. Find the dimensions of the dog run.

Example 6 Using Ratios

The lengths of the sides of a triangle are in the ratio 3:4:5. The perimeter of the triangle is 18 in. Find the lengths of the sides.

The lengths of the sides of a triangle are in the ratio 12:13:15. The perimeter of the triangle is 120 cm. Find the lengths of the sides.

Example 7 Real World Connection

Radar detected an unidentified plane 5000 mi away, approaching at 700 mi/h. Fifteen minutes later an interceptor plane was dispatched, traveling at 800 mi/h. How long did the interceptor take to reach the approaching plane?

1-4 Solving Inequalities

1.03 Operate with algebraic expressions (polynomial, rational, complex fractions) to solve problems.

What you’ll learn … To solve and graph

inequalities To solve and write

compound inequalities

Properties of Inequalities

Let a, b and c represent real numbers. Transitive Property If a≤b and b≤c, then a≤c. Addition Property If a≤b, then a+c ≤b+c. Subtraction Property If a≤b, then a-c ≤b-c. Multiplication Property If a≤b and c>0, then ac ≤bc. If a≤b and c<0, then ac ≥bc.

Division Property If a≤b and c>0, then a/c ≤b/c. If a≤b and c<0, then a/c ≥b/c.

Solving Inequalities

• To solve an inequality, use the same procedure as solving an equation with one exception. When multiplying or dividing by a negative number, reverse the direction of the inequality sign.

• To graph the solution set, circle the boundary and shade according to the inequality.

• Use an open circle for < or > and closed circles for ≤ or ≥.

Example 1 Solving and Graphing Inequalities

3x – 12 < 3 6 + 5(2 - x) ≤ 41

Example 2 No Solutions or All Real Numbers as Solutions

2x – 3 > 2(x – 5) 4(x – 3)+ 7 ≥ 4x + 1

Real World Connection

The band shown at the left agrees to play for $200 plus 25% of the ticket sales. Find the ticket sales needed for the band to receive at least $500

Compound Inequalities

A compound inequality is a pair of inequalities that are joined by the words “and” or “or”.

Example 4 Compound Inequality Containing And

3x -1 > -28 and 2x +7 < 19

2x >x +6 and x -7 ≤ 2

Example 4 Compound Inequality Containing Or

4y -2 ≥ 14 or 3y - 4 ≤ -13

X – 1 < 3 or x +3 > 8

Real World Connection

The plans for a gear assembly specify a length of 13.48 cm with a tolerance of +0.03 cm. A machinist finds that the part is now 13.67 cm long. By how much should the machinist decrease the length?

1-5 Absolute Value Equations and Inequalities

2.08 Use equations and inequalities with absolute value to model and solve problems; justify results. a) Solve using tables, graphs, and algebraic properties.

Objective 1: Absolute Value Equations

The absolute value of a number is its distance from 0 on a number line and distance is nonnegative.

3 3

3 3

Example 1 Solving Absolute Value Equations

2y – 4 = 12 3x + 2 = 12

Example 2 Solving Multi-Step Absolute Value Equations

3 4w – 1 - 5 = 10 2 3x - 1 + 5 = 33

Example 3 Checking for Extraneous Solutions

2x + 5 = 3x + 4 x = x - 1

An extraneous solution is an answer that is NOT a solution

Example 4 Solving Inequalities of the form A ≥ b

3x + 6 ≥ 12 2x – 3 > 7

Properties

Let k represent a positive real number.

x ≥ k is equivalent to x ≤ -k or x ≥ k.

x ≤ k is equivalent to -k ≤ x ≤ k.

Example 5 Solving Inequalities of the form A < b

3 2x + 6 - 9 < 15 5z + 3 - 7 < 34

Real World Connection

The specifications for the circumference C in inches of a basketball for men is 29.5 ≤ C ≤ 30. Write the specification as an absolute value inequality.

In Chapter 1, You Should Have

Reviewed and extended your knowledge of algebraic expressions and your skills in solving equations and inequalities.

Solved absolute value equations and inequalities by changing them to compound equations and inequalities.