Chapter 5 Notes
Lesson 5.1: Perimeter and Area
1
2
Lesson 5.2: Solving Equations with Variables on Each Side
3
Lesson 5.3: Inequalities
Example 1 Write an Inequality
Write an inequality for each sentence.
a. The account balance is more than $50.
Words
The account balance
Symbols
Let b represent account balance.
Inequality
is more than
$50.
b > 50
b. The temperature is less than or equal to 34.
Words
The temperature is less than or equal to
Symbols
Let t represent temperature.
34.
t  34
Inequality
Example 2 Determine Truth of an Inequality
For the given value, state whether each inequality is true or false.
a. 3 > 4a – 9; a = 2
3 > 4a – 9
Write the inequality.
?
3  4(2) – 9
3 > -1
Replace a with 2.
Simplify.
This sentence is true.
x
- 2  8; x = 120
6
x
Write the inequality.
6-28
?
120
2
8
Replace x with 120.

6
18  8
Simplify.
This sentence is false since 18  8.
Example 3 Graph an Inequality
b.
4
Graph
each
inequality on a number line.
a. x < 4
4
Locate 4 on the number line. It is a
key point in the inequality.
0 1 2 3 4 5 6 7 8 9 10
4
o
0 1 2 3 4 5 6 7 8 9 10
4
o
0 1 2 3 4 5 6 7 8 9 10
Draw an open dot on 4. An open dot
is used when the number is not
included.
The inequality x < 4 means that all
numbers less than 4 will make the
sentence true. Show that by drawing
a line from the dot with an arrow
pointing to the left.
b. x  -3
–3
Locate -3 on the number line. It is a
key point in the inequality.
-7 -6 -5 -4 -3 -2 -1 0 1 2 3
–3

-7 -6 -5 -4 -3 -2 -1 0 1 2 3
–3

-7 -6 -5 -4 -3 -2 -1 0 1 2 3
Draw a closed dot on -3. A closed dot
is used when the number is included.
The inequality x  -3 means that all
numbers greater than or equal to -3
will make the sentence true. Show
that by drawing a line from the dot
with an arrow pointing to the right.
Example 4 Write an Inequality
Write an inequality for the graph.
A closed circle is on 3, so the point 3 is included in the graph. The arrow points to the left, so
the graph includes all numbers less than or equal to 3. The inequality is x  3.
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Lesson 5.4: Solving Inequalities
Example 2 Graph the Solution of an Inequality
2
Solve w - > 4. Graph the solution on a number line.
5
2
w–5>4
Write the inequality.
2 2
2
w-5+5>4+5
Addition Property of Inequality
2
w > 45
Simplify.
2
The solution is w > 45. Check your solution.
Since the inequality symbol is >,
Graph the solution.
2
45
2
draw an open dot at 4 and a
5
line with an arrow to the right.
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Example 3 Multiply or Divide by a Negative Number
Solve each inequality and check your solution. Then graph the solution on a number
line.
a. –2x  10
-2x  10
Write the inequality.
 2x
10

Division Property of Inequality
2
2
x  -5
Simplify. Check the result.
Graph the solution x  -5.
p
b. – < -1
4
p
–4 < -1
 p
-4    > -4(-1)
 4
p>4
Write the inequality.
Multiplication Property of Inequality
Simplify. Check the result.
Graph the solution p > 4.
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Lesson 5.5: Solving Multi-Step Equations and Inequalities
Example 1 Solve Equations and Inequalities with Parentheses
a. Solve 4(a – 3) = 8. Check your solution.
4(a – 3) = 8
Write the equation.
4a - 12 = 8
Use the Distributive Property.
4a - 12 + 12 = 8 + 12
Addition Property of Equality
4a = 20
Simplify.
4a 20
4 = 4
Division Property of Equality
a=5
b. Solve 9  2(t + 5).
9  2(t + 5)
9  2t + 10
9 – 10  2t + 10 – 10
–1  2t
Simplify.
Write the inequality.
Use the Distributive Property.
Subtraction Property of Inequalities
Simplify.
–1 2t
2 2
1
1
–  t or t  –
2
2
Division Property of Inequalities
Simplify.
Example 2 Solve Multi-Step Equations
Solve 6(y – 3) = 4(6 + y). Check your solution.
6(y – 3) = 4(6 + y)
Write the equation.
6y – 18 = 24 + 4y
6y – 4y – 18 = 24 + 4y – 4y
2y - 18 = 24
2y – 18 + 18 = 24 + 18
2y = 42
2y
42
=
2
2
y = 21
Use the Distributive Property.
Subtraction Property of Equality
Simplify.
Addition Property of Equality
Simplify.
Division Property of Equality
Check your solution.
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Example 3 Solve Multi-Step Inequalities
Solve –3(k − 8) < k – 4. Graph the solution on a number line.
–3(k − 8) < k – 4
Write the inequality.
–3k + 24 < k – 4
Use the Distributive Property.
–3k + 24 + 3k < k – 4 + 3k
Addition Property of Inequality
24 < 4k – 4
24 + 4 < 4k – 4 + 4
28 < 4k
Simplify.
Addition Property of Inequality
Simplify.
7 < k or k > 7
o
Division Property of Inequality
Graph the solution on a number line.
0 1 2 3 4 5 6 7 8 9 10
Example 4 Null Set and Identities
a. Solve 4x – 9 = 2(2x + 1).
4x – 9 = 2(2x + 1)
Write the equation.
4x – 9 = 4x + 2
Use the Distributive Property.
4x – 4x – 9 = 4x – 4x + 2
-9 = 2
Subtraction Property of Equality
Simplify.
The statement –9 = 2 is never true. The equation has no solutions and the solution set
is Ø.
b. Solve 12x – 8 = 3(4x – 3) + 1.
12x – 8 = 3(4x – 3) + 1
Write the equation.
12x – 8 = 12x – 9 + 1
Use the Distributive Property.
12x – 8 = 12x – 8
Simplify.
12x – 8 + 8 = 12x – 8 + 8
12x = 12x
x=x
Addition Property of Equality
Simplify.
Division Property of Equality
The statement x = x is always true. The equation is an identity and the solution set is all
numbers.
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