Name: __________________________
Algebra 1-1: Lesson 13
Goal – Solve and graph equations and inequalities with variables on both sides.
END GOAL FOR SOLVING ALL EQUATIONS:
Get a variable by itself on one side of the =.
TOOLS:
 Distributive property
 Combine like terms on one side
 Inverse operations to combine like terms on opposite sides
TIP:
Solve ALL equations the
same way—think about
your goal and your tools
to get there.
These are the only things you
need for solving equations!!!
Hi! Styracosaurus here. So you know how we’ve been doing
inverse operations to “cancel things out?” Well, we’re going to take
a look at what that actually means. Fill in the missing number in
each equation by actually doing the operation to both sides.
Addition/Subtraction Equations
Addition
Subtraction
𝑥 + 5 = 12
𝑥 − 5 = 12
−5 − 5
+5 + 5
𝑥 + ____ = 7
𝑥 + ____ = 17
𝑥=7
𝑥 = 17
What number do you get when you
“cancel” with addition/subtraction?
EXAMPLE 1
Multiplication/Division Equations
Multiplication
Division
𝑥
= 18
3𝑥 = 18
3
1
÷3 ÷3
∙ 𝑥 = 18
3
∙
3
∙3
____ ∙ 𝑥 = 6
𝑥=6
____ ∙ 𝑥 = 54
𝑥 = 54
What number do you get when you
“cancel” with multiplication/division?
Solve an equation with variables on both sides
Solve the equation, if possible.
a.
b. 3 + 4𝑚 = 4𝑚
13 − 6𝑥 = 3𝑥 − 14
1
c. 4𝑥 − 7 = (9𝑥 − 15)
d. 4(4𝑥 − 5) = 2(8𝑥 − 10)
3
Solution:
End goal: get a variable by itself on one side of the =
a.
13 − 6𝑥 = 3𝑥 − 14
−3𝑥 − 3𝑥
.
13 − 9𝑥 = −14
−13
− 13
−9𝑥 = −27
÷ −9 ÷ −9
𝑥=3
Draw a wall through the middle of the =.
Do inverse operations in SADMEP order to get the variables on the
same side of the =.
Identify the #s and operations on the same side of the = as the variable.
Do inverse operations in SADMEP order.
 Answer!
~~Lesson 13: Solve and graph equations and inequalities with variables on both sides.~~
Draw a wall through the middle of the =.
b. 3 + 4𝑚 = 4𝑚
−4𝑚 − 4𝑚
Do inverse operations in SADMEP order to get the variables on the
same side of the =.
3=0
All the variables cancel, so check to see if the statement is true or false.
DIRTY, DIRTY LIES!
There is nothing that we
could plug in for m to make
this true, so the answer is:
No Solution.
 Answer!
Draw a wall through the middle of the =.
Distribute through the ( ).
1
c. 4𝑥 − 7 = 3 (9𝑥 − 15)
4𝑥 − 7 = 3𝑥 − 5
−3𝑥
− 3𝑥
Do inverse operations in SADMEP order to get the variables on the
same side of the =.
Identify the #s and operations on the same side of the = as the variable.
Do inverse operations in SADMEP order.
.
1𝑥 − 7 = −5
+7 + 7
 Answer!
𝑥=2
d. 4(4𝑥 − 5) = 2(8𝑥 − 10)
16𝑥 − 20 = 16𝑥 − 20
−16𝑥
− 16𝑥
.
−20 = −20 
Draw a wall through the middle of the =.
Distribute through the ( )s.
Do inverse operations in SADMEP order to get the variables on the
same side of the =.
All the variables cancel, so check to see if the statement is true or false.
TRUE STORY!
This is always true, so we can
plug in any number for x, so
the answer is:
Infinitely Many Solutions.
 Answer!
Exercises for Example 1
Solve the equation, if possible.
1. 9𝑎 = 7𝑎 − 8
2. −5𝑐 + 6 = 9 − 4𝑐
3. 4(3𝑥 − 2) = 2(6𝑥 + 1)
4. 11𝑥 + 7 = 10𝑥 − 8
5. 5(3𝑥 − 2) = 3(5𝑥 − 1)
6.
1
(6𝑥
2
+ 18) = 3(𝑥 + 3)
~~Lesson 13: Solve and graph equations and inequalities with variables on both sides.~~
EXAMPLE 2
a.
Solve and graph an inequality with variables on both sides
Solve the inequality, if possible, then graph the solution.
9𝑥 + 2 < 5𝑥 − 18 b. 12𝑥 − 3 > 2(6𝑥 − 15)
c. 24𝑎 − 22 < −4(1 − 6𝑎) d. 2(4𝑥 − 3) − 8 ≤ 4 + 2𝑥
Solution:
End goal: get a variable by itself on one side of the inequality sign and graph with shading
Draw a wall through the middle of the <.
a. 9𝑥 + 2 < 5𝑥 − 18
−5𝑥
− 5𝑥
Do inverse operations in SADMEP order to get the variables on the
same side of the <.
4𝑥 + 2 < −18
Identify the #s and operations on the same side of the < as the
−2
−2
variable.
4𝑥 < −20
Do inverse operations in SADMEP order.
÷4 ÷4
Graph the inequality using an open circle because it is not equal to.
𝑥 < −5
.
–7
–6
–5
–4
–3
Draw a wall through the middle of the >.
Distribute through the ( )s.
b. 12𝑥 − 3 > 2(6𝑥 − 15)
12𝑥 − 3 > 12𝑥 − 30
−12𝑥
− 12𝑥
Do inverse operations in SADMEP order to get the variables on the
same side of the >.
.
−3 > −30
TRUE STORY!
Infinitely Many Solutions and you shade in
the whole graph.
–2
–1
0
1
2
All the variables cancel, so check to see if the statement is true or
false.
You cannot graph this inequality since there are no numbers that
will make the statement true.
DIRTY, DIRTY LIES!
No Solution and you don’t shade in
anything.
0
1
2
Combine like terms on the same side of the ≤.
Do inverse operations in SADMEP order to get the variables on the
same side of the ≤.
8𝑥 − 14 ≤ 4 + 2𝑥
−2𝑥
− 2𝑥
6𝑥 − 14 ≤ 4
+14 + 14
2
 Answer!
Draw a wall through the middle of the ≤.
Distribute through the ( ).
d. 2(4𝑥 − 3) − 8 ≤ 4 + 2𝑥
8𝑥 − 6 − 8 ≤ 4 + 2𝑥
1
 Answer!
Do inverse operations in SADMEP order to get the variables on the
same side of the <.
−4 < −4
–1
All the variables cancel, so check to see if the statement is true or
false.
Graph the inequality by shading in the entire number line since any
number at all would work.
Draw a wall through the middle of the <.
Distribute through the ( )s.
c. 24𝑎 − 4 < −4(1 − 6𝑎)
24𝑎 − 4 < −4 + 24𝑎
−24𝑎
− 24𝑎
–2
 Answer!
6𝑥 ≤ 18
÷6 ÷6
Identify the #s and operations on the same side of the ≤ as the
variable.
Do inverse operations in SADMEP order.
𝑥≤3
Graph the inequality using a closed circle because it is equal to.
3
4
5
 Answer!
~~Lesson 13: Solve and graph equations and inequalities with variables on both sides.~~
HEY STYRACOSAURUS!
Wait a minute, the numbers are different in b. but
the same in c., so shouldn’t the answers be reversed?
GOOD THINKING, T-REX!
Don’t think about it as looking for the “same numbers” or “different numbers.”
You want to think about it as looking at the whole statement and figuring out whether
or not it’s true. So since −3 IS bigger than −30, the statement is true. And
since −4 is NOT less than itself, the statement is false.
Exercises for Example 2
Solve the inequality, if possible, then graph the solution.
7. 10 − 3𝑥 ≤ 5𝑥 − 14
8.
9. 𝑝 − 4 < −9 + 𝑝
1
(8𝑥
2
1
+ 6) < 3 (9𝑥 − 15)
10. −18 − 6𝑘 < 6(1 − 𝑘)
1. What is this section about?
2. What is one important idea in this section?
3. What is one mistake that someone might make when trying to do these kinds of problems?
~~Lesson 13: Solve and graph equations and inequalities with variables on both sides.~~
Extra Practice:
You must complete the circled problems before you take your knowledge celebration.
You may complete as many of the others as you need to feel comfortable with the material.
Solve the equation or inequality, if possible. Graph the inequalities.
1. −7𝑎 + 9 = 3𝑎 + 49
2. 4(𝑤 + 3) = 𝑤 − 15
3. 8(𝑦 − 5) = 6𝑦 − 18
4. 8𝑥 − 2 = −9 + 8𝑥
5. 8𝑚 − 7 < 4𝑚 + 5
6. 10 − 11𝑑 > −5𝑑 − 4
7. 9𝑧 ≤ −7𝑧 + 14
8. 5𝑝 − 14 = 8𝑝 + 4
~~Lesson 13: Solve and graph equations and inequalities with variables on both sides.~~
3
9. 2(𝑥 − 3) − 8 = 4 + 2𝑥
10. (−8𝑥 − 2) ≥ −4 − 8𝑥
2
11. 2𝑥 + 8 ≤ 2𝑥 − 2
12. 6𝑤 + 3 < 2𝑤 + 15
13. 2 − 𝑥 > 3𝑥 + 10
14. 𝑎 + 5 = −5𝑎 + 5
15. 14𝑚 − 10 = 3(4 + 𝑚)
16. 7 + 𝑥 = (4𝑥 − 2)
1
2
~~Lesson 13: Solve and graph equations and inequalities with variables on both sides.~~
17. 8𝑏 + 11 − 3𝑏 = 2𝑏 + 2
18. 10𝑑 − 6 > 4𝑑 − 15 − 3𝑑
19. 16𝑝 − 4 = (2𝑝 − 3)
20. 0.25(8𝑧 − 4) = 𝑧 + 8 − 2𝑧
Find the perimeter of the square. (Hint: You’ll have to find x first, THEN find the perimeter.)
21.
22.
23.
~~Lesson 13: Solve and graph equations and inequalities with variables on both sides.~~
Was the problem completed correctly? If not, find and correct the error.
25. 3𝑛 − 5 = −8(6 + 5𝑛)
24. 𝑝 − 1 > 5𝑝 − 8 − 4𝑝
𝑝−1>𝑝−8
−𝑝
−𝑝
.
−1 > −8
No solution, since they’re different numbers.
3𝑛 − 5 = −8(11𝑛)
3𝑛 − 5 = −88𝑛
−3𝑛
− 3𝑛
−5 = −91𝑛
÷ −91 ÷ −91
5
91
=𝑛
Pick a number that you can put in the blank and rewrite the equation…
a) To make the equation have No Solution and
b) To make the equation have Infinitely Many Solutions.
27. ____𝑥 + 12 = 2(5𝑥 + 6)
26. 4𝑥 + ____ = 4𝑥 + 7
a)
a)
b)
b)
28. 9𝑚 − 4 = −3𝑚 + 5 + ____𝑚
29. 2𝑥 + 7 = −1(____ − 2𝑥)
a)
a)
b)
b)