9.4 – Problem Solving General Guidelines for Problem Solving 1. Understand the problem. Read the problem carefully. Identify the unknown and select a variable.

9.4 – Problem SolvingGeneral Guidelines for Problem Solving

1. Understand the problem.Read the problem carefully.

Identify the unknown and select a variable.

Construct a drawing if necessary.

2. Translate the information to an equation.

3. Solve the equation and check the solution.

4. Interpret the solution.

Example 1:

Three times the difference of a number and 5 is the same as twice the number decreased by 3. Find the number.

k is the number

5k 3 2k 3

3 5 2 3k k

Three times

The difference of a number and 5

istwice the number

decreased by 3

9.4 – Problem Solving

Example 1:

3 5 2 3k k

3 15 2 3k k

13 15 2 35 15k k

3 2 12k k

12k

3 2 22 12k kk k

9.4 – Problem Solving

Example 1:

3 5 2 312 12

3 7 24 3

21 21

Check:

9.4 – Problem Solving

Example 2:

The difference between two positive integers is 42. One integer is three times as great as the other. Find the integers.

x = one integer

3x x 42

3 42x x

The difference between two positive integers

is 42

3x = the other integer

9.4 – Problem Solving

Example 2:

3 42x x

2 42x

2 42

2 2

x

21x

23 3 631x

21 23 421

Check:

63 21 42

42 42

9.4 – Problem Solving

Example 3:

A 22-foot pipe is cut into two pieces. The shorter piece is 7 feet shorter than the longer piece. What is the length of the longer piece?

Longer piece = m

7m m 22

7 22m m

Longer piece Shorter pieceplus is 22 feet

Shorter piece = m – 7

9.4 – Problem Solving

Example 3:7 22m m

2 7 22m

77 22 72m

2 29m

14.5m feet

2 29

2 2

m

9.4 – Problem Solving

Example 3:

14.5 14. 25 7 2

29 7 22

22 22

Check:

9.4 – Problem Solving

Example 4:

A college graduating class is made up of 450 students. There are 206 more females than males. How many males are in the class?

Males = h

206h h 450

206 450h h

Males Femalesplus is 450 students

Females = h + 206

9.4 – Problem Solving

Example 4:

206 450h h

2 206 450h

2062 2206 0 0645h

2 244h

122h males

2 244

2 2

h

9.4 – Problem Solving

Example 4:

122 122 206 450

244 206 450

450 450

Check:

9.4 – Problem Solving

Example 5:

A triangle has three angles A, B, and C. Angle C is 18 degrees greater than angle B. Angle A is 4 times angle B. What is the measure of each angle?

4 18 180B B B

180m A m B m C

18m C B 4m A B m B B

m A plus m B m C is 180plus

18018B B4B

9.4 – Problem Solving

Reminder:The sum of the angles in a triangle is:

Example 5:

4 18 180B B B

6 18 180B

186 18 81 180B

6 162B

27B

6 162

6 6

B

27m B

9.4 – Problem Solving

Example 5:Check:Other angles:

274m A 108m A

2 87 1m C 45m C

180m A m B m C

108 27 45 180

180 180

9.4 – Problem Solving

9.5 – Formulas and Problem SolvingGeneral Guidelines for Solving for a Specific Variable

in a Formula1. Eliminate fractions from the formula.

2. Remove parentheses from the formula using the distributive property.

3. Simplify like terms.

4. Get all terms containing the specified variable on one side of the equation.

5. Use the multiplicative inverse property to get the specified variable’s coefficient to one.

6. Simplify the results if necessary.

Example 1:

9, 63d rt t d

63 9r

63 9

9 9

r

7 r

Using the given values, solve for the variable in each formula that was not assigned a value.

7r

Check:

63 9r

63 97

63 63

9.5 – Formulas and Problem Solving

Example 2: Volume of a Pyramid1

40, 83

V Bh V h

40 81

3B 1

40 83

3 3 B

120 8B

15 B

120 8

8 8

B

15B

9.5 – Formulas and Problem Solving

LCD: 3

Example 2: Volume of a Pyramid1

40, 83

V Bh V h

Check:

0 851

43

1

40 5 8

40 40

9.5 – Formulas and Problem Solving

Example 3: Solve for the requested variable.

1

22 2A bh

1

2A bh

Area of a Triangle – solve for b

2A bh

2A bh

h h

2Ab

h

9.5 – Formulas and Problem Solving

LCD: 2

Example 4: Solve for the requested variable.

932

532 32F C 9

325

F C

Celsius to Fahrenheit – solve for C

932

5F C

5 32 9F C

5 32

9

FC

932

55 5F C

5 32 9

9 9

F C

or 532

9F C

9.5 – Formulas and Problem Solving

LCD: 5

Example 4: Solve for the requested variable.

932

532 32F C 9

325

F C

Celsius to Fahrenheit – solve for C

932

5F C 9

325

5 5

9 9F C

532

9F C

Alternate Solution

9.5 – Formulas and Problem Solving

Guidelines for Using Formulas in Problem Solving

1. Understand the problem.Read the problem carefully.Identify the known, unknown and the variable(s).Construct a drawing if necessary.

2. Translate the information to a known formula.

3. Solve the equation and check the solution.

4. Interpret the solution.

Formulas describe a known relationship among variables. Most formulas are given as equations, so the guidelines for problem

solving are relatively the same.

9.5 – Formulas and Problem Solving

Example 1:

A pizza shop offers a 2-foot diameter round pizza and a 1.8-foot square pizza for the same price. Which one is the better deal?

Round Pizza Square Pizza2Area r 2.Area Sq s

2d 1r 3.14

23.14 1Area

23.14Area ft

1.8s

2. 1.8Area Sq

2. 3.24Area Sq ft

9.5 – Formulas and Problem Solving

Example 2:

A certain species of fish requires 1.6 cubic feet of water per fish. What is the maximum number of fish that could be put into a tank that is 3 feet long by 2.4 feet wide by 2 feet deep?

1.6 f l w h

Cubic feet is a unit of volume.Volume for Fish Volume of Tank

Number of fish (f)

timesRequired

volume per fish

equals length*width*height

f * 1.6 = 3*2.4*2

1.6 3 2.4 2f

9.5 – Formulas and Problem Solving

Example 2:

1.6 f l w h 1.6 3 2.4 2f 1.6 7.2 2f

1.6 14.4f

1.6 14.4

1.6 1.6

f

9f fish

9.5 – Formulas and Problem Solving

9.6 - Linear Inequalities and Problem SolvingProperties of Inequality

Addition Property of Inequality

If a, b, and c are real numbers, then

and ora b a c b c anda b a c b c

(The property is also true for subtraction.)

Properties of Inequality

Multiplication Property of Inequality

1. If a, b, and c are real numbers and c is positive, then

anda b ac bc

anda b ac bc

2. If a, b, and c are real numbers and c is negative, then

are equivalent inequalities.

are equivalent inequalities.

9.6 - Linear Inequalities and Problem Solving

Graphing an Inequality

4y

2x

5 1x

0 1 2 3 4 5 6-1-2-3-4-5-6

0 1 2 3 4 5 6-1-2-3-4-5-6

0 1 2 3 4 5 6-1-2-3-4-5-6

9.6 - Linear Inequalities and Problem Solving

Guidelines for Solving a Linear Inequality1. Eliminate fractions from the formula.

2. Remove parentheses from the formula using the distributive property.

3. Simplify like terms.

4. Get all terms containing the specified variable on one side of the equation using the addition property of inequality.

5. Use the multiplication property of inequality to get the specified variable’s coefficient to one.

6. Simplify the results if necessary.

9.6 - Linear Inequalities and Problem Solving

*****Reverse the inequality sign when multiplying or dividing by a negative value.*****

Solve each inequality and graph the solution.

7 127 7x 7 12x

5x

Example 1:

0 1 2 3 4 5 6-1-2-3-4-5-6

9.6 - Linear Inequalities and Problem Solving

Solve each inequality and graph the solution.

8 7 10 47 7x x 8 7 10 4x x Example 2:

0 1 2 3 4 5 6-1-2-3-4-5-6

8 10 3x x 8 1010 310x xx x

2 3x 2 3

2 2

x

3

2x 1.5x

9.6 - Linear Inequalities and Problem Solving

Solve each inequality and graph the solution.

18 218 124 83x x

18 2 3 24x x

6x

Example 3:

0 1 2 3 4 5 6-1-2-3-4-5-6

2 3 6x x 32 3 63x xx x

9.6 - Linear Inequalities and Problem Solving

Solve each inequality and graph the solution.

8 12 3

21 7x x

Example 4:

8 1221 21 3

21 7x x

8 2 3 3x x 8 16 3 3x x

8 16 3 9x x 18 16 3 96 16x x 8 3 7x x

9.6 - Linear Inequalities and Problem Solving

LCD: 21

Example 4:

0 1 2 3 4 5 6-1-2-3-4-5-6

1.4x

5 7

5 5

x 7

5x

8 33 3 7x xx x 5 7x

8 3 7x x

9.6 - Linear Inequalities and Problem Solving