Post on 23-Dec-2015
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 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 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 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