Algebra 2
Chapter 1 Notes
1.4 Solving Equations
Topics:
Expectations:
Solving Equations
Translating Words into Algebra
Solving Word Problems
A: Solving One-Variable Equations
The equations below are easy types of equations that you
solved often in Algebra I.
One Step
x+6=9
Two Step
1.4 Solving Equations
-3y = 15
I want to see for any problem:
The original problem
Any key steps in getting to your
solution- “the work”
Clearly stated solution
Answers:
Should use original variable if
applicable x = 2 or y = 5, etc.
FRACTIONS should always be
reduced to lowest terms.
DECIMALS only if they are
terminating and you write the entire
thing… never round unless the
directions say so.
2a  3  6
In Algebra II, you will face some equations that are bit more challenging to solve.
Example 1: Solve each of the following. Show your work!
1a.
6x  5  7  9x
1b.
5
3 2
v 
6
8 3
*Get like
denominators OR
use a scientific
calculator with a
fraction button*
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Algebra 2
Chapter 1 Notes
1c.
5(6  4 y)  y  21
1e.
x 1 
1
 x  2
2
1d.
1.4 Solving Equations
53 = 3(y − 2) − (3y – 1)
1f.
B: Translating Algebra to Words and back (review lesson 1.3)
TIPS:
“A number” means the
variable… pick one! I like
“x” and “n”, but any letter
will do!
Try to pick out words that
mean operators (+, ×, ÷, −)
“is” means “=”
Use ( ) to group a “sum” (etc)
together.
Example 2:
Write an algebraic expression to represent each verbal
expression.
2a. three less than a number
2b. six times the cube of a number is the quotient of the same
number and 81
2c. the square of a number decreased by the product of the same
number and five
2d. twice the difference of a number and six is equal to 24
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Algebra 2
Chapter 1 Notes
Example 3:
Write a verbal expression to represent each equation.
3a.
3b.
3c.
1.4 Solving Equations
TIPS:
6=−5+x
7y − 2 = 19
m
 3(2m  1)
5
Read the expression out
loud… that is what you
should write down.
Write “a number” for any
variable
A group in parenthesis is
usually a “sum” “difference”
“product” or “quotient”
You must SPELL the first
word of any sentence even if
it is a number. After that,
you may use numerals.
C: Word Problems
Procedure for solving word problem
1. Relax. Read through the problem at least twice.
2. Identify a variable. Call it whatever you want, x, n, etc. and tell what it
means/stands for (example: let x = the 5th test score)
3. Write an equation that contains that variable by translating the English words
into Math symbols and numbers
4. Solve the equation.
5. Answer the question asked and make sure your answer makes sense
Solve each of the following story problems. Be sure to name a variable, show your
equation and your work, and answer the question being asked in words.
1.
a) The sum of three consecutive integers is 78. What are the integers?
b) How would you change the variables for “Three consecutive odd integers”?
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Algebra 2
Chapter 1 Notes
1.4 Solving Equations
2.
Sam’s scores on his first four tests are 84, 75, 86, and 92.
What must he score on the fifth test to average 88%?
3.
Tickets to A Midsummer Night’s Dream cost $10.50 for adults and $7.50 for students. Mrs. Smith
ordered $192 worth of tickets for a field trip for her English class. If a total of 24 tickets were
ordered, how many of each type of ticket did she order?
4.
The length of a rectangle is twice as long as the width. Find the dimensions of the rectangle if
the perimeter is 360 ft.
5.
The sides of a triangle are in a ratio of 4:5:6 . The perimeter of the triangle is 52.5 inches. Find
the lengths of each side.
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Algebra 2
Chapter 1 Notes
1.5 Solving Inequalities
1.5 Solving Inequalities
A: Writing, Solving, and Graphing Inequalities
Inequality Summary

greater than



less than
greater than or equal
less than or equal
Inequalities represent values that may not necessarily be equal.
Example: What inequality represents:
“Fourteen minus 5 times a number is no more than 20” ?
You may recall from Algebra 1, that to graph an inequality in one variable, you use a closed or open dot
on the number and then shade to the left or right.
Solving inequalities is the same as solving equations EXCEPT if you multiply or divide by a negative
number, you have to FLIP the inequality symbol.
Solve the example from above, then graph the solution.
Examples 1:
1a.
Solve each inequality. Graph the solution.
1b.
1c.
5
Algebra 2
Chapter 1 Notes
1d. **
1.5 Solving Inequalities
1e.**
B: Solving Compound Inequalities
There are two types of compound inequalities: The “AND” and the “OR”
Type 1: “AND” Compound Inequalities
A B
“A” and “B” means the overlapping values common to A and B.
Example 2: Solve each compound inequality. Graph the solution.
2a.
3x  12 and 8x  16
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Algebra 2
Chapter 1 Notes
1.5 Solving Inequalities
2b.
3x  4  16 and 2 x 1  13
2c.
13  2x  7  17
2d.
19  3 y  2  10
2e.
5x  15 and
Type 2: “OR” Compound Inequalities
“A” or “B” means the all the area covered by A and all area covered by B
x6 8
A B
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Algebra 2
Chapter 1 Notes
1.5 Solving Inequalities
Example 3: Solve each compound inequality. Graph the solution.
3a. y  2  3 or y  4  3
3c.
3b.
4d 1  9 or 2d  5  11
3x  9 or 2 x  10
Example 4:
4a. A farmer wants to make a rectangular pen using no more than 400 ft. of fencing. If he wants the
length to be exactly 10 feet longer than the width, describe the possible dimensions of the pen.
Write an inequality and solve.
4b. Write an inequality for the following situation: “This medicine should be stored between 65°F and
75°F for optimum life.”
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Algebra 2
Chapter 1 Notes
1.6 Absolute Value Equations and Inequalities
1.6 Solving Absolute Value Equations and Inequalities
A: Solving Absolute Value Equations
Perhaps you remember from a previous math class the concept of “absolute value.”
Solve this equation:
Strategy:
x A
Set up 2 equations:
x  A or x   A
1a.
1c.
x 5
x  18  5
1
x  5  10
3
Example 1: Solve the following absolute value equation. Be sure to
check your answers.
The universal symbol for “no solution” is 
1b.
a  9  30
1d.
5 2 x  4  7  17
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Algebra 2
Chapter 1 Notes
1.6 Absolute Value Equations and Inequalities
1e.
5x  6  9  0
1f.
1h.
6 5  2 y  9
1g.
5  3 2  2w  7
x  6  3x  2
B: Solving Absolute Value Inequalities
List the possible integer values of x that make
x 5
What do you notice?
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Algebra 2
Chapter 1 Notes
1.6 Absolute Value Equations and Inequalities
There are two types of absolute value inequalities: “Less than” and “Greater than”
“GO L.A.!”can help you remember the difference
Greater than = Rewrite and solve like an Or inequality.
Less than = Rewrite and solve like an And inequality.
Example 2:
Solve each inequality. Graph the solution.
2a.
4 x  8  20
2b.
3x 12  6
2c.
4s  1  27
2d.
10  2k  2
11
Algebra 2
2e.
3b  5  2
Chapter 1 Notes
1.6 Absolute Value Equations and Inequalities
Example 3:
Example 4:
A chemist is preparing a solution that must contain 3 grams of acid. The acid tolerance
for the solution is described by the inequality a  3  0.005 . What is the weakest acid solution allowed?
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