Chapter 3: Solving Linear Equations & Proportions
Examples
Things to Remember & Study

Solving Equations
Example:
1
2
1
2
x = 2(3 + x)
Use distributive prop.
x = 6 + 2x
-2x +
1
2

“Move” the 2x term
x=6
Combine the x terms
3
−2 x = 6
Eliminate the fraction
3
2(− 2) x = 2(6)
-3x = 12
-3
Eliminate the coefficient
-3
x = -4

Solving Proportions
Example:
5
=
13
𝑘−4
39
cross-multiply:
5
=
13
5(39) = 13(𝑘 − 4)
52 + 195 = 13𝑘
247
13𝑘
=
13
13
19 = 𝑘
19−4
39
39
A proportion is an equation that states two
ratios (fractions) are equivalent.
Solve a proportion by cross-multiplying to
create a new equation; then solve the
equation.
Use distributive property if the numerator or
denominator is an expression.
Pg 144-145: 3-35 odd (answers in yellow section
SA5)
Pg 150-151: 3-31 odd (answers in yellow section
SA6)
http://www.math-play.com/Two-Step-EquationsGame.html
http://www.mathplayground.com/AlgebraEquation
s.html
http://www.quia.com/cb/77775.html
Pg 171: 3-29: odd (answers in yellow section
SA6)
http://www.arcademics.com/games/dirt-bikeproportions/dirt-bike-proportions.html
http://www.arcademics.com/games/dirt-bikeproportions/dirt-bike-proportions.html
https://jeopardylabs.com/play/ratios-unit-rateand-proportions
247 = 13𝑘
5


195 = 13𝑘 − 52
Check: 13 =
𝑘−4
To solve an equation for a particular variable
means to get that variable by itself with a
coefficient of 1 on one side of the equation
Rules for manipulating equations
- Use distributive property to eliminate ( )
- “Move” term from one side of equation to
the other side (move the term and change
the sign)
- Combine like terms on one side
- Eliminate a fraction coefficient by
multiplying EVERY term on BOTH sides by
the denominator
- Eliminate an integer coefficient by dividing
EVERY term on BOTH sides by that
coefficient
For Extra Help & Practice
=
15
39
=
5
13

Chapter 3: Solving Linear Equations & Proportions
Try These While Looking at Examples
Answers
Try These Without Looking
1) 6 = –7f + 4f
1) –2 = f
1) 21 = n
2) 12v + 14 + 10v = 80
2) v = 3
2) –5 = z
3) 27 = 3c – 3(6 – 2c)
3) 5 = c
3) m = 3
4) 5 = y
4) x = 1
5) NO.
5) NO.
5(10) – 3(10 – 6) =
1
(2(6) – 10) =
2
1
50 – 3(4) =
1
2
2
4) 4 = 9 (4y – 2)
5) A student solved this equation:
5x – 3(x – 6) = 2
and got an answer of x = 10.
Is this answer correct?
50 – 12 =
3
1) 𝑦 =
6
15
6
42
77
3) 4+2𝑤 =
(2) =
(2x – 10) = 4
and got an answer of x = 6.
1) y = 7
1) x = 14
Solve the proportion.
2) t = 7
2) n = 12
1) 3 =
3) w = 7
3) c = 22
4) 5 cups of flour
4) A recipe that yields 12 buttermilk
biscuits calls for 2 cups of flour. How
much flour is needed to make 30
biscuits?
3
4) 2 (x – 5) = –6
Is this answer correct?
2
2)
4) 18 minutes
−2
𝑤−13
3) 5m + 2(m + 1) = 23
5) A student solved this equation:
35
2) 𝑡+4 =
2) 9 = 7z – 13z - 21
1≠4
38 ≠ 2
Solve the proportion.
2
2
1) 10 = 7 n + 4
12
30
=
2
𝑓
7.2
𝑚
=
8
20
12𝑓 = 60
144 = 8𝑚
𝑓=5
18 = 𝑚
3)
𝑛−2
50
𝑐−8
−2
𝑥
21
=
=
6
30
11−4𝑐
11
4) It took 7.2 minutes to upload 8 digital
photographs from your computer to a
website. At this rate, how long will it take
to upload 20 photographs?
Chapter 6: Solve & Graph Inequalities
Examples
Things to Remember & Study
x<3

<--------------------0------------------3--------->
x > -1
The sign “points” to the smaller value and
“eats” the larger value
 < : “is less than”
For Extra Help & Practice
Pg. 359: 3-21 odd (answers in yellow section
SA16)

<---------- –1--------0--------------------------->
x ≤ -1

<--------- –1 --------0-------------------------->
x≥3

<-----------------0-------------------- 3 -------->
-1 < x ≤ 3 (x > -1 AND x ≤ 3)


<--------- –1 -----0------------------- 3 ------->

> : “is greater than”

≤ : “is less than or equal to”
 ≥ : “is greater than or equal to”
When graphing, use  (open circle) for < & >
Use  (closed circle) for ≤ & ≥
WATCH OUT! for combinations.
“AND” means that the graph will have one line
with 2 endpoints.
x ≤ -1 OR x > 3

“OR” means the graph will have arrows pointing
in opposite directions.
Solve the inequality.

To solve an inequality, follow the same
rules as solving equations, EXCEPT you
SWITCH THE INEQUALITY SIGN when:
- you multiply by a NEGATIVE number
- you divide by a NEGATIVE number


Some inequalities have NO solutions
Some inequalities have a solution of ALL
REAL NUMBERS

<---------- –1 ----0------------------ 3 ------->
–3(w + 12) > 0
–3w – 36 > 0
move -36
–3w > 36
divide by –3
w < –12
SWITCH sign
–5 > 10
always FALSE, no solution
–5 < 10
always TRUE, all real #s
Pg 372: 3-27 odd (answers in yellow section
SA16)
Chapter 6: Solve & Graph Inequalities
Try These While Looking at Examples
Answers
Try These Without Looking
Write the inequality for the graph.

1) <----------------0----------------- 5 -------->
1) x < 5
2) x ≥ -3
3) x > -2 AND x ≤ 4
or
-2 < x ≤ 4

2) <------- –3 -------0----------------------->
4)

<----------0------------ 2 -------------->

7) <--------------0------------------- 8 --->
5)


<----------0------------ 3 -------------->
Write the inequality for the graph.


3) <----- –2 -----0------------------- 4 ------->
6) x > -2
Graph the inequality.
7) x ≤ 8
4) x > 2
8) x < 2
9)
<----------------------0------------------------->
5) x ≤ 0 OR x > 3
OR x ≥ 7
≤≥
1) -11m ≤ -22
2) 2(t – 3) > 2t – 8
3) 2(s + 4) ≤ 16
9) x ≥ -7



<---- -3 ----------------0---------- 2 ------>
<----------------------0------------------------->
Solve the inequality.


8) <----------0------ 2 -------------- 7 ------->
Graph the inequality.
<---- -7 -------------0-------------------->
10)

6) <-------- -2 -------0------------------------->
<----------------------0------------------------->
10) x ≤ 2 AND x ≥ -3
<----------------------0------------------------->
1)
2)
3)
4)
5)
6)
m≥2
no solution
s≤6
v > 0.6
all real numbers
x≤5
Solve the inequality.
≤≥
4) 8.2 + v > -7.6
5) 12x – 1 > 6(2x – 1)
6) 3x – 7 ≤ 8