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Mrs. Bondi - Pre-Algebra 2085
Solving Inequalities Class Notes
Lesson 1 Inequalities and Their Graphs (Glencoe text 7.3)
Objective:
to graph inequalities with one variable
Equation – a mathematical sentence that uses an ___________ sign to show that the two expressions have
the __________ value
Inequality – a mathematical sentence that uses an inequality symbol/sign to _____________ the values of
two expressions
Use a representative arrow on a number line to visually indicate the values that make the inequality true.
Equality Symbol
Term
Graphing Symbol
equal to

Term
Graphing Symbol
<
less than

>
greater than

≤
less than or equal to

≥
greater than or equal to

≠
not equal to

=
Inequality Symbols
Be sure the variable is on the left side of the inequality before graphing. (It will be far easier!)
Graph each equality or inequality.
x=1
x≠1
x≤1
x<1
x≥1
x>1
0≤x≤2
0<x<2
A solution to an inequality is any value that makes the inequality true. Every value darkened on your graph
is a solution to the inequality.
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Mrs. Bondi - Pre-Algebra 2085
Solving Inequalities Class Notes
Examples:
1) Determine whether each number is a solution of the given inequality.
2x + 4 < 20
a. 2
b. 10
Graph each inequality.
2. n ≥ 5
3. j > –4
4. k ≤ 10
5. m < -3/2
Write an inequality for each graph.
6.
7.
Define a variable and write an inequality to model each situation.
8. No more than 10 people may use the treadmills at any time in the gym.
9. To train for a marathon, a runner decides that she must run at least 12 miles each day.
Practice:
HW: p.343 #14-44 even
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Mrs. Bondi - Pre-Algebra 2085
Solving Inequalities Class Notes
Lesson 2 Solving Inequalities (Glencoe text 7.4)
Objective:
to solve and graph inequalities with one variable using addition or subtraction
Solution of the Inequality – any value(s) of the variable that makes the inequality true
Example: For x > 10, the solution is all numbers greater than 10.
To solve an inequality, follow the same process used to solve equalities.
When adding and subtracting, the process works exactly the same as solving an equality.
Example:
x+7=6
x+7<6
–7 –7
–7 –7
x = –1
x < –1
Set Notation: {x: x = –1}
{x: x < –1}
Set Builder Notation – demonstrates parts needed to build desired set – {x: x < -1}
Remember, it is easiest if you are sure the variable is on the left side of the inequality before graphing.
Examples: Solve and graph each inequality. Write your solution in set notation.
1)
r  6  22
2)

3
g  4  4
Mrs. Bondi - Pre-Algebra 2085
Solving Inequalities Class Notes
3)
n   4  7
4)
c  3  40
6)
k 1111

5)
x90


7)
A family earns at most $2500 a month. The family’s expenses are $2000. Write and solve an
inequality to find the possible amounts of money the family could deposit into a savings account each month.
Practice: Write your solution in set notation.
HW: p.348 #22-46 even, 51-56
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Mrs. Bondi - Pre-Algebra 2085
Solving Inequalities Class Notes
Lesson 3 Solving Inequalities (Glencoe text 7.5)
Objective:
to solve and graph inequalities with one variable using multiplication or division
To solve an inequality, follow the same process used to solve equalities.
When multiplying and dividing by a positive number, the process works exactly the same as with an
equation.
Example:
7x = 42
7
7
x=6
7x < 42
7
7
x<6
BUT … When multiplying or dividing by a negative number, REVERSE THE SIGN of the inequality.
Example:
–7x = 42
–7 –7
x = –6
-7x < 42
–7 –7
x > –6
When multiplying or dividing by a negative number
REVERSE THE SIGN of the inequality
Remember, it is easiest if you are sure the variable is on the left side of the inequality before graphing.
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Mrs. Bondi - Pre-Algebra 2085
Solving Inequalities Class Notes
Examples: Solve and graph each inequality. Write each solution in set notation.
**** Remember to reverse the sign of the inequality if you Multiply OR Divide by a Negative!
1)
4q  48
2)
 7 p  21
3)
3q  12
4)
-10y ≤ -60
5)
x
12
3
6)
x
 12
3




7)
x
 12
3
8)
2
m2
7

Define a variable and write an inequality to solve these problems.
9) You wonder if you can save money by using your cell phone for all long distance calls. Long distance calls
cost $.05 per minute on your cell phone. The basic plan for your cell phone is $29.99 each month. The cost of
regular phone service with unlimited long distance is $39.99. Define a variable and write an inequality that will
help you find the number of long-distance call minutes you may make and still save money.
11)
10) The unit cost for a piece of fabric is $4.99 per
yard. You have $30 to spend on material. How many
feet of material could you buy?
HW: p.353 #14-38 even, 39, 42, 47
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Mrs. Bondi - Pre-Algebra 2085
Solving Inequalities Class Notes
Lesson 4 Solving Multi-Step Inequalities (Glencoe text 7.6)
Objective:
to solve multi-step inequalities and graphing the solutions on a number line
You solve a multi-step inequality in the same way you solve a one-step inequality. You use the properties of
inequality to transform the original inequality into a series of simpler, equivalent inequalities.
Just like when solving multi-step equalities, we need to isolate the variable.
First, on each side of the inequality, distribute and/or combine like terms if needed.
Second, move all variable terms to the left side of the inequality.
Third, move anything added or subtracted to the variable to the other side using the inverse operation.
Fourth, move anything multiplied to or dividing the variable to the other side using the inverse operation.
Remember, when multiplying or dividing by a negative number, REVERSE THE SIGN of the inequality.
Examples: Solve each inequality and graph your solution.
1)
4a  3  10
2)
3x  4  6  x   2
3)
3w  6  3  2w 1
4)
2( w  2)  3w  1
5)
4(k + 2) – 3k ≤ 12
6)
3(2c – 2) – 2c > 0
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Mrs. Bondi - Pre-Algebra 2085
Solving Inequalities Class Notes
7) A grandmother devises an inequality to help her remember the ages of her two grandchildren. She
knows her grandson is two years older than her granddaughter and that together they are at least 12
years old. What are the youngest that her grandson and granddaughter could be?
6) A family decides to rent a boat for the day. The boat’s rental rate is $500 for the first two hours and
$50 for each additional half hour. Suppose the family budgeted $700 to rent the boat. What is the
maximum number of additional half hours for which they can rent the boat?
Practice:
HW: p.357 #18-34 even, 37-40
Good Idea: Study Guide and Review p.360-362;
Practice Test p.363
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