Solving One-Step Equations

Solution of an equation or\ç\__Q

Equivalent equations:

Properties of Equality:Let a, b, and c represent real numbers.

AddionProperty:

ft+c6c Ex. 5hQSubtraction Property:

Ex.

Multiplication Property:Ex.

Division Property:

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SubstitufionProp

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2b ÷ 3bInverse operations:‘

S+th, MR D V S lQj\Example 1: Solving Using Addition or Subtraction

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b. g+7=11a. x—3=—o

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Example 2: Solving Using Multiplication or Division

aZLx=9 b. —96=4c3 9

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Section 3.1 — Solving Two-Step Equations

Steps to follow for solving two-step equations:‘ ‘Mo GJ’ -kcj\ O rc&(thQ’\2. c oW\cco o

Example 1: Solving a Two-Step Equation

a. Solve 10=+2 b. x—3=2 CkQ- 4_• 6

Example IA music store sells a copy of deluxe electric guitar for $295. This is $30 more than -i-the cost ofthe deluxe electric guitar it is modeled after. What is the cost of the deluxe electric guitar?

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Example 3: Writing a FunctionIn a catalog, tulips cost $0.75 each and shipping costs are $3.00. Write a rule that describes theamount spent as a function of the number of bulbs ordered. Then determine the greatest numberof bulbs that you can order for $14.

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Example 4: Using Deductive Reasoning 4Z 9a. Solve 1 — +5 and justify each step.12

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I ccb. Solve 9—b =11 and justify each step.

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Section 3.2 Solving Multi-Step Equations

Steps for solving a multi-step equation:

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Example 1: Combining Like Terms

Solve each equation.a. 2c+c+12=78 b. —4b+16--2b=46

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z3orExample 2: Real-World ConnectionA gardener is planning a rectangular garden area in a community garden. His garden will benext to an existing 12-ft fence. The gardener has a total of 44 ft of fencing to build the otherthree sides of the garden. How long will the garden be if the width is 12 ft?

—(6L(l -4(Q-j(. k(4

ii ft

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i+i

= X cçExample 3: Solving an Equation with Grouping Symbols

a. Solve —2(b—4)=12

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b. Solve —31 = —(4x —5) + 3x —6

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Example 4: Solving an Equation That Contains Fractions

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ExampleS: Solving an Equation That

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b. Solve.5 2

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b. Solve —2.3x+8+1.43x=—3.68

— 2x xa. Solve —+—=7

2’3 2

4x 7Qo

riti?Decimals

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44o\a. Solve O.5a+8.75= 13.25

+5=35-21i S75

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Section 3.3 -‘ Equations With Variables on Both Sides

Example 1: Solving an Equation With Variables on Both Sides

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a. 7x—5=2x+15-2c ‘\

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Example 2: Vertical AnglesFind the value of x in the diagram below.

b. —(x+6)+5x+8=2—2x

-x2-2x

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-

3=x-akX

Example 3: Real-World ConnectionYou can buy used in-line skates from your friend for $40, or you can rent some. Either way, youmust rent safety equipment. How many hours must you skate for the cost of renting and buyingskates to be the same?

3=X ocjj

SKATE RENTALSLu-line skates sixdsafeti’ eeulpment

$3.50/hourSafety equipment

$1.50/hour

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Example 4: Solving Using a Graphing Calculator3 1

Solve m 8— m using a graphing calculator.

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1denti:

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( [Uk oWNo solution:

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Example 4: Identities and Equations With No Solutions

a. Solve 10— 8a = 2(5 —4a) b. Solve 6rn — 5= 7rn+ 7—rn

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Section 3.4 Ratio and Proportion

Ratio: Ccpcco dsc, C1cRate: b Ckc ct\t (std- s dc’), --4’i 1\J

0 SUnit Rate:i I tU 0

Example 1: Using Unit RatesThe table at the left gives prices for different sizes of the same brand of apple juice. Find theunit rate (cost per ounce) for the I 6-oz size.

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I Price of Apple Juice

Price I Voiwne 1L_$72Ii6J

-- S; \v:f(V;z— 0.Example 2: Real-World Connection

In 2004, Lance Armstrong won the Tour de France, completing the 3391 km course in about83.6 hours. Find Lance’s unit rate, which is his average speed. Write a rule to describe thedistance he cycles d as a function of the time t he cycles. Cycling at his average speed, abouthow long it would take Lance to cycle 185 km?

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$120 32oz ii,J! ‘\ç’

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Example 3: Converting Rates

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— Iii-.ic\r,A cheetah ran 300 feet in 2.92 seconds. What was the cheetal’i din miles perhour?

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Proportion:o -Q9uoJ rck

c CV:ICaCExample 4: Using the Multiplication Property of Equality

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Cross Products of a Proportion:

Cb = Cross

o Soe o 0poi+o’ 4 kZZ5ExampleS: Using Cross Products LA box of cereal weighing 354 grams contains 20 grams of fat. Find the number of grams of fatin the recommended serving size of 55 grams.

35- 55

9S &-i* 20 -x

Example 6: Solving Multi-Step PiOj,’rtions

x+4 x—2Solve the proportion= 7

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-c

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2’-\’ : iDilation:

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Scale Factor: —--

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Example 2: Dilating Figures on the Coordinate PlaneQuadrilateral PQRS has vertices P(-2, 4),Q(4, 4), R(4,-2), and S(-4,-4). It is dilated by a scalefactor of, and the origin is the center of dilation. Graph the original figure and its dilation.

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Section 3.5 Proportions and Similar Figures

Similarfigures:CQS 1nO* CJQ *hQ iI\’& [ypQiZxi-- CQ rIat\c sc

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— CO1Q oicsExample 1: Finding the Length f a Side ‘

AABCADEF.FindDE.

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Example 3: Applying SimilarityA tree casts a shadow 7.5 ft long. A woman 5 ft tall casts a shadow 3 ft long. The triangle shownfor the tree and its shadow is similar to the triangle shown for the woman and her shadow. Howtall is the tree?

Scale drawing: (\-n QrooQc sc T’O- S_

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Example 4: Finding Distan4es on MapsThe scale of the map at the left is I inch&10 miles. Approximately how far is it from Valkaria toWabasso?

Section 3.6 — Equations and Problem SoIvin

Example 1: Defining One Variable in Terms of Another

The length of a rectangle is 6 in. more than its width. The perimeter of the rectangle is 24in. What is the length of the rectangle?

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Consecutive Integers:r

Example 2: Consecutive Integer Problem

The sum of three consecutive integers is 147. Find the integers.

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Example 3: Same-Direction Travel

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b. Define a variable:

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c. Complete the table:

d. Set up an equation and solve:

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A train leaves a train station at 1 P.M. It travels at an average rate of 72 mi/h. A high-speed train leaves the same station an hour later. It travels at an average rate of 90 mi/h.The second train follows the same route as the first train on a track parallel to the first. Inhow many hours will the second train catch up with the first train?

a. Draw a diagram:‘ ,.-\cc\ --

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Example 4: Round-Trip TravelNoya drives into the city to buy a software program at a computer store. Because of trafficconditions, she averages only 15 mi/h. On her drive home she averages 35 mi/h. If the totaltravel time is 2 hours, how long does it take her to drive to the computer store?

a. Draw a diagram:

6 bfs.b. Define a variable:

+o 9 ± t GIQa -t CC5 +0 9Q+c. Complete the table:

Rate Time DistanceTo the computer store

b 5tHome from thecomputer store ( a 3

d. Set up an equation and solve:

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Example 5: Opposite-Direction TravelJane and Peter leave their home traveling in opposite directions on a straight road. Peter drives15 mi/h faster than Jane. After 3 hours, they are 225 miles apart. Find Peter’s rate and Jane’srate.

a. Draw a diagram: —

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b. Define a variable:

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c. Complete the table:

d. Set up an equation and solve:

3r3r+Lf5= /V Fck)

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Rate Time DistanceJane

Peter3 3tS;i+4

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Section 3.7 Percent of Change

Percent of Change:,-

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Percent of Increase:

)mo)r-tPercent fDecrease:

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Example 1: Finding Percent of Change

The price of a sweater decreased from $29.99 to $24.49. Find the percent of decrease.

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Example 2: Real-World Connection

In 1990, there were 1330 registered alpacas in the United States. By the summer of 2000,there were 29,856. What is the percent of increase in registered alpacas?

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Greatest Possible Error: /

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-3aIrrational Numbers: ) 1

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)Example 2: Rational and Irrational Square Roots s—’ —S.

Tell whether each expression is rational or irrational.

a. 49 b. -Ji

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c.=

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Example 3: Estimating Square Roots

Between what two consecutive integers is 114.52?

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9 So +hQ\Example 4: Approximating Square Roots With a Calculator \Yi or 3Find g14.52 to the nearest hundredth.

Example 5: Real-World Connection

The formula d = Jx2 + (2x)2 gives the length d of each wire for the tower below. Findthe length of the wire ifx = 12 ft.

d

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x

Example 3: Finding the Greatest Possible Error

You use a beam balance to find the mass of a rock sample for a science lab. You read thescale as 3.8 g. What is your greatest possible error?

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Example 4: Finding Maximum and Minimum Areas

You measure a room and make the diagram below. Use the greatest possible error to findthe maximum and minimum possible areas.

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Percent Error: —\ no. roQcà -Th

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Example 5: Finding Percent Error

Suppose you measure a CD and record its diameter as 12.1 cm. Find the percent error inyour measurement.

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Example 6: Finding Percent Error in Calculating Volume

The diagram below shows the dimensions of a cassette case. Find the percent error incalculating its volume.

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Section 3.9 - The Pythagorean Theorem

Parts of a Right Triangle:

Hypotenuse:

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Leg:*c\

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Example 2: Real-World Connection

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A fire truck parks beside a building such that the base of the ladder is 16 ft from thebuilding. The fire truck extends its ladder 30 ft as shown below. How high is the top ofthe ladder above the ground?

The Pythagorean Theorem:In any right triangle, the sum of the square of the lengths of the legs is equal to the

square of the length of the hypotenuse.

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Example 1: Using the Pythagorean Theorem

What is the length of the hypotenuse of the triangle below?

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1 (ilL

1 ÷cH c’C2-

900

Conditional:

t ftHypothesis:

Conclusion: Q ±h.

Converse: cocsmn a cccd oncd.

The Converse of the Pythagorean Theorem:

If a triangle has sides of lengths a, b, and c, and a2 + = c2, then the triangle is aright triangle with hypotenuse of length c.

Example 3: Using the Converse of the Pythagorean Theorem

Determine whether the given lengths can be sides of a right triangle.

a. 5in.,l2in.,andl3in. b. 7m,9m,andl2m

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