1. 2
Chapter
Chapter 2
Equations, Inequalities and Problem Solving

2. An Introduction to Problem Solving
Section
2.4
An Introduction to Problem Solving

3. Solving Direct Translation Problems
Objective 1
Solving Direct Translation Problems

4. Writing Phrases as Algebraic Expressions
Addition
( + )
Subtraction
( - )
Multiplication
( ∙ )
Division
( ÷ )
Equal Sign
Sum
Difference of
Product
Quotient
Equals
Plus
Minus
Times
Divide
Gives
Added to
subtracted from
Multiply
Into
Is/was/ should be
More than
Less than
Twice
Ratio
Yields
Increased by
Decreased by Of
Divided by
Amounts to
Total
Less
Represents
Is the same as
Objective A

5. Strategy for Problem Solving

6. Example Twice a number plus 3 is the same as the number minus 6.
Objective A

7. Example
The product of twice a number and three is the same as the difference of five times the number and ¾. Find the number.
1. Understand
Read and reread the problem.
If we let x = the unknown number, then
“twice a number” translates to 2x,
“the product of twice a number and three” translates to 2x · 3,
“five times the number” translates to 5x, and
“the difference of five times the number and ¾” translates to
5x – ¾.
Continued

8. Example (cont) 2. Translate The product of · the difference of –
is the same as =
twice a number 2x
5 times the number 5x
and
and 3 3
Continued

9. Example (cont) 3. Solve 2x · 3 = 5x – ¾ 6x = 5x – ¾
4. Interpret
Check: Replace “number” in the original statement of the problem with –¾. The product of twice –¾ and 3 is 2(–¾)(3) = –4.5. The difference of five times –¾ and ¾ is 5(–¾) –¾ = – We get the same results for both portions.
State: The number is –¾.

10. Solving Problems Involving Relationships Among Unknown Quantities
Objective 2
Solving Problems Involving Relationships Among Unknown Quantities

11. Example
A car rental agency advertised renting a Buick Century for $24.95 per day and $0.29 per mile. If you rent this car for 2 days, how many whole miles can you drive on a $100 budget?
x = the number of whole miles driven, then
0.29x = the cost for mileage driven
2(24.95) x = 100
Continued

12. Example (cont) 2(24.95) + 0.29x = 100 49.90 + 0.29x = 100
Continued

13. Example (cont)
Check: Recall that the original statement of the problem asked for a “whole number” of miles. If we replace “number of miles” in the problem with 173, then (173) = , which is over our budget. However, (172) = 99.78, which is within the budget.
State: The maximum number of whole number miles is 172.

14. Example
The measure of the second angle of a triangle is twice the measure of the smallest angle. The measure of the third angle of the triangle is three times the measure of the smallest angle. Find the measures of the angles.
Draw a diagram.
Let x = degree measure of smallest angle
2x = degree measure of second angle
3x = degree measure of third angle
Continued

15. Example (cont)
Recall that the sum of the measures of the angles of a triangle equals 180.
measure of second angle
measure of third angle
measure of first angle
180
equals x 2x + 3x + =
180
Continued

16. Example (cont) x + 2x + 3x = 180 6x = 180 x = 30 Check:
If x = 30, then 2x = 2(30) = 60 and 3x = 3(30) = 90
The sum of the angles is = 180.
State:
The smallest angle is 30º, the second angle is 60º, and the third angle is 90º.

17. Solving Consecutive Integer Problems
Objective 3
Solving Consecutive Integer Problems

18. Example
The sum of three consecutive even integers is 252. Find the integers.
x = the first even integer
x + 2 = next even integer
x + 4 = next even integer
Translate: x + x x + 4 = 252
Continued

19. Example (cont)
The sum of three consecutive even integers is 252. Find the integers.
x + x x + 4 = 252
3x + 6 = 252
3x = 246
The integers are 82, 84 and 86.