Chapter 2
Intermediate Algebra
1
One Variable Equations
(Mathematical) Expression –
(Mathematical) Equation –
Roots / Solutions –
Some equations have __________ answer, some have ___________ answers, some have more
than 2 answers and some have __________________ answers while other have ____________
answer.
Linear Equation –
Solve the following for x:
1 1

12 2
x 3  7
x
2
 x6
3
6x  8  3x  7
Homework
Print rest of packet
Page 65 - 66 16, 21, 23, 24
31, 41, 51
3x  4  6
3(x  2)  3  2(6  x)
Chapter 2
3x 9

2 2
3x  2 x x
 
12
9 2
Intermediate Algebra
4x 5

3 2
x 7 x 1
  
2 8 6 4
3x 
2x  1 5

4
3
3
5
1
x  x 3  x 5
2
6
3
Homework
Page 67 67, 71, 77, 78, 81, 83, 93
Quiz
2
Chapter 2
Intermediate Algebra
3
Word Problems
Name an integer
Name an even integer
Name an odd integer
Name 3 consecutive integers
Name 3 consecutive even integers
Name 3 consecutive odd integers
If x is an integer, what is the next integer? What is the next next integer?
If x is an even integer, what is the next integer? What is the next, next even integer?
If x is an odd integer, what is the next odd integer? What is the next, next odd integer?
The sum of three consecutive integers is 78, Find the integers.
Define variable
Write equation
Solve equation
Answer question
The difference of two times an odd integer and 1/3 the next odd interger is 41. Find the two
consecutive odd integers.
Chapter 2
Intermediate Algebra
One integer is 4 more than another integer. The sum of those two integers is 26. Find the two
integers.
Homework
Page 75 28, 29, 31, 26
If you have 8 nickles then you have __________________ dollars, because:
If you have “n” nickels then you have _________________ dollars because:
If you have “d” dimes and “q” quarters:
How may coins do you have?
How much money, in dollars, do you have?
You have a total of 22 coins. You only have dimes and quarters. You have $4.45. How many
quarters doyou have?
4
Chapter 2
Intermediate Algebra
At the end of the day the cash register has as many $5 bills as $10 bills. Those bills total $2400.
How many of each bill is there?
A collection of stamps contains 3cent, 10 cent and 15 cent stamps. The number of 10 cent stamps
is two more than twice the number of 3 cent stamps. There are three times as many 15 cent
stamps as there are 3 cent stamps. The total value of the stamps is $1.56. Find the number of 15
cent stamps.
Homework
Page 73 5, 7, 9, 12
If you run for 2 hours at the rate of 8 mph, you will run _______________ miles because:
If you run for 40 minutes at the rate of 8 mph, you will run ________________ miles because:
If you drove your car 110 miles in 3 hours, on average, how fast were you driving?
5
Chapter 2
Intermediate Algebra
One bicyclist pedals at the rate of 15 mph, another, in the opposite direction, at the rate of 10
mph. If they started at the same time from the same place, when will they be 10 miles apart?
A boat can travel at the rate of 16 mph, but it is traveling up river against a 4 mph current. How
far will the boat travel in 2 hours?
A plane can fly at the rate of 300 mph. If flies with a tail wind and makes the 850 mile trip in 2.5
hours. How fast is the tail wind?
Homework
Page 85-6 21, 24, 30, 33
6
Chapter 2
Intermediate Algebra
Write “3 percent” in three different ways:
Simple Interest –
Compound Interest –
12% yearly, compounded monthy will yield
%
You invest $2000 at a 3% simple annual interest rate. At the end of one year how much money
did you earn? How much money do you now have in total?
Interest earned (I) =
You invested money in a 2.5% simple interest account and earned $45. How much did you
invest?
7
Chapter 2
Intermediate Algebra
You have $8000 to invest. You put some in a 5.4% account and the rest in a 7.2% account. You
earned a total of $553.50. How much did you invest in each account?
Homework
Page 92 5, 6, 7, 13
Inequalities
x = 4 has how many answers?
x > 4 has how many answers?
How can we represent (show) those answers?
-
-
-
Except for one special case, solving an inequality is like solving an ____________________,
you want to isolate, get by itself, the variable.
8
Chapter 2
Intermediate Algebra
Solve for x and show the answer three ways:
3x – 4 < 2x – 1
x + 3 ≥ 4x
-3x > 9
5(x- 2) ≥ 9x – 3(2x -4)
Homework
Page 103 9, 12, 13, 25, 31, 35
9
Chapter 2
Intermediate Algebra
Solving Compound Inequalities
Type I “ or ”
2x + 3 > 7 or 4x – 1 ≤ 3
a) solve each inequality separately
b) place answers on one number line and deteremine final answer.
Note: For “or” problems, every part of the number line that has at least one of the lines
drawn on it is part of the final answer.
c) write answer in interval notation.
2x + 3 > 7 or 4x – 1 ≤ 3
If each of the following represent the solutions to a compound inequality, determeine the final
answer and write it using simplest interval notation.
x < 5 or x > 7
x > 5 and x < 7
x < 5 or x < 7
x > 5 and x > 7
10
Chapter 2
x > 5 or x > 7
Intermediate Algebra
x < 5 and x < 7
x > 5 or x ≤ 7
x < 5 and x ≥ 7
x < 5 or x ≤ 7
x > 5 and x ≥ 7
Solve for x, write answer in simplest interval notation:
3x – 4 > 5 or x – 3 < -2
11
Chapter 2
Intermediate Algebra
12
For each of the following compound inequalities identify which, if any, of the following list of
numbers is (are) part of the solution.
-3 -2.5 -2 -1.5 -1 -.5 0 .5 1 1.5 2 2.5 3 3.5 4 4.5 5
-2 < x < 3
x ≤ -2
or x > 3
x ≥ -2 and x > 3
Type II 3-sided inequalities
1 < 3x – 5 < 4
a) get the variable by itself in the middle part of the inequality by doing the same thing to all
three sides.
b) If done correctly and completely the result of step a should be the answer in one symbol
notation.
c) represent the answer in simplest interval notation.
1 < 3x – 5 < 4
-2 ≤ 5x + 3 ≤ 13
Homework
Page 105 55, 59, 69, 71
Chapter 2
Intermediate Algebra
13
Type III “ and “
11 – 2x > -3 and 7 – 3x < 4
a) solve each inequality separately
b) place answers on one number line and deteremine final answer.
Note: For “and” problems, any part of the number line that has both of the lines drawn
on it, is part of the final answer.
c) write answer in simplest interval notation.
11 – 2x > -3 and 7 – 3x < 4
5 – 4x > 1
and 6 – 5x < 11
Homework
Page 105 53, 61, 65, 75, 81
Absolute Value Equations
Remember that absolte value, | |, measuers the distance from the value to 0 on a number line.
|3|=
What is x if | x | = 3 ?
|-3|=
What is x if | x | = -3?
Chapter 2
Intermediate Algebra
To solve an absolute value equation:
1. Using algebra get the | | by itself on one side of the equation.
2. Create two equations joined by the word “or”:
One equation is what is in the | | set equal to the other side.
The other is what is in the | | set equal to the negative of the other side.
Note:These two equations should not have |
| symbols in them.
3. Solve each equation
|x+2|=8
| 2 - x | = 12
| 2x | = -4
| 2x – 1 | = 0
3 - | 2x – 4| = -5
3|x – 1| = 9
Page 114
Homework
8, 11, 16, 17, 22, 26, 37, 40
14
Chapter 2
Intermediate Algebra
Absolute Value Inequations
Solve :
|x|<5
Remember the definition of absloulte value.
Type I “less than”
| 4x – 3| < 5
To solve | | < equations:
1. Using algebra get | | by itself on one side.
2. Create a 3-sided inequality following this pattern:
negative of the “other side” < what is inside the | | < the “other side”
Note: This inequality should not have |
| symbols in it.
When creating this inequality always use < or ≤ symbols
3. Write the answer using interval notation.
Solve:
| 4x – 3| < 5
| 3 – 7x| ≤ 17
15
Chapter 2
Intermediate Algebra
Type II “greater than”
| 2x – 1 | ≥ 7
To solve | | > equations:
1. Using algebra get | | by itself on one side.
2. Create two inequalities joined by the word “or”:
For one inequality simply remove the | | symbols.
For the other also remove the | | symbols but switch the symbol and the sign of the “other
side”.
Note: These inequaliies should not have |
| symbols in it.
3. Write the answer using interval notation.
| 2x – 1 | ≥ 7
| 5x + 3 | > 3
Homework
Page 114-115 55, 56, 57, 59, 63, 73
16
Chapter 2
Intermediate Algebra
Special Case for Absolute Values
| x |=0
| x | < 0
| x | > 0
| x | = -2
| x | < -2
| x | > -2
Homework
Page 115 66, 67, 69, 71, 73, 75, 76
17