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 numbers To 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 0.31 Irrational Numbers 5 8 10 -3 Integers -10 5 2 3 Whole Numbers 4/2 - 123 6 25 -2 3 -4 0.37 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? The cost C of admission for n people The maximum speed s in meters per second on a roller coaster of height h in meters The park’s profit (or loss) P in dollars for each week w of the year

Example 2: Graphing Numbers on a Number Line Change to Decimals - 3 1.7 5 2

Example 3: Ordering Real Numbers An inequality is a mathematical sentence that compares the value of two expressions using an inequality symbol. > Greater than < Less than > Greater than or equal to < Less than or equal to Compare - 0.25 and - 0.01

Example 4: Finding Inverses The reciprocal or multiplicative inverse of any nonzero number a is 1/a. The product of reciprocals is 1. Find reciprocal of -3.2 400 1/5 The opposite or additive inverse of any number a is –a. The sum of opposites is 0. Find the opposite of -3.2 400 1/5

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 Distributive a(b+c) = ab+ac

Example 5: Identify Properties of Real Numbers Which property is illustrated? 6+(-6) = 0 (-4 1) - 2 = -4 – 2 (3+0) - 5 = 3 – 5 -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.

x + y ÷ x c2 - d2 3x – 4y + x – y c(3 – d) – c2 x + 2x ÷ y - 2y 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 3c d y 6c – 2d 2x d 2x – y 2d 3x 3c 8c +d 6c – 2d 5x – 2y

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 Example 4 Solving an Equation for One of Its Variables 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 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 What you’ll learn … To solve and graph inequalities To solve and write compound inequalities 1.03 Operate with algebraic expressions (polynomial, rational, complex fractions) to solve problems.

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

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 An extraneous solution is an answer that is NOT a solution 2x + 5 = 3x + 4 x = x - 1

Example 4 Solving Inequalities of the form A ≥ b

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 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.