Topic 2: Solving Equations
& Inequalities in One
Variable
Algebra 1
Table of Contents
1.
2.
3.
4.
5.
6.
7.
8.
Solving Multi-Step Equations Advanced
Special Types of Equations
Literal Equations
Graphing Inequalities
Solving Inequalities
Solving Compound Inequalities
Absolute Value Equations & Inequalities
Equations & Word Problems
Solving MultiStep Equations
Advanced
How to solve Multi-Step Equations
1. Distribute.
Bonus Step: Multiple by reciprocal/LCM, if fractions.
2. Combine Like Terms (ONLY on same side of
equal sign).
3. Use the inverse operation to move numbers
to the right.
4. Use the inverse operation to move variables
to the left.
5. Divide.
Example
2
(4x
3
– 8) = 16
Example
6(4 + 3x) – 10(x – 5) = 30 – 2(x + 4)
Let’s Practice…
1. 4(5 + 2x) + 7x = -5(x + 7)
Let’s Practice…
2.
2
𝑥
3
+6=−
5
(4𝑥
2
− 2)
Special Types of
Equations
No Solution…
• Happens when variables cancel each other
out…
• Example:
10x + 17 = 2(5x +3) + 15
Identity
• Happens when the simplified equations are
identical.
• Example:
9m + 4 = - 3m + 4 + 12m
Technically, there are infinitely many solutions.
Let’s Practice…
Solve the following equations. If there is a special
situation, identify what type (no solution or
identity).
1. 5y + 2 =
1
(10y
2
+ 4)
2. 2(2k - 1) = 4(k – 2)
Let’s Practice…
Solve the following equations. If there is a special
situation, identify what type (no solution or
identity).
3. 4 – d = -(d - 4)
4. 5(7 – x) + 12 =
1
(10 𝑥
2
+ 8)
Literal
Equations
What is a Literal Equation?
Literal Equation: an equation that involves 2 or
more variables.
Solving Literal Equations
• You solve literal equations just like regular
equations!
Examples
Solve:
1. ax – b =c for x
2. A =
𝑏ℎ
2
for b
Let’s Practice
Solve:
1. mx + 2nx = p for x
2. a(8 + n) = 17 for n
Example
Let’s Practice
Graphing
Inequalities
What is an Inequality?!
Inequality: statement that two
quantities are not equal.
Symbols of comparison:
<
>
≤
≥
≠
Graphing Inequalities
Remember:
- If you take the time to
draw the line, you take
the time to fill it in!
t>4
2
3
4
5
6
7
8
y  11
9
10
11
12
13
14
15
- The inequality sign
(when the variable is on
the left) always points in
the same direction as
the arrow on the
number line.
Let’s Practice…
• Graph the following inequalities:
1. 5 ≥ x
2. x < 7
3. x ≤ -3
Example
Create and graph an inequality:
In order to pass the Algebra 1 EOC, you must
earn at least 399 points on the exam.
Let’s Practice…
Define a variable and write an inequality to
model each situation.
1. The school auditorium can seat at most 1200
people.
2. For a certain swim meet, a competitor must
swim faster than 23 seconds to qualify.
Solving
Inequalities
Solving Inequalities
We solve inequalities, just like regular
algebraic expressions with equal signs (=).
Just pretend the inequality is an equal
sign!!!
Example
5x + 10 ≥ 3x + 12
Special Negative Rule:
If you divide by a negative, you must flip
the inequality symbol.
For example:
-9w < 6 + 3w
Example
8x + 12 ≤ 28 + 16x
Let’s Practice…
Solve and graph each inequality.
1. t + 10 > 4 + 3t
2. - 6m ≤ 150
Let’s Practice…
Solve and graph each inequality.
3. 4(p + 2) - 10 > 14 + 3p
𝑥
3
4. < 10 – 7x
Let’s Practice…
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? Write an
inequality to model this situation and then
solve.
Solving
Compound
Inequalities
What is a Compound Inequality?
• A compound inequality consist of two distinct
inequalities joined by the word “and” or the
word “or”.
– Example: All real numbers greater than -3 and less
than 2.
How to Solve a Compound
Inequalities
Step 1: Separate
into separate
inequalities.
Step 2: Solve.
Step 3: Combine
solutions
Step 4: Graph
each solution on
the same
number line.
Example
- 3 ≤ m – 4 < -1
Let’s Practice…
Solve and graph each inequality.
1. 3 < 2p  3  12
2. 6b  1  41 or
2b + 1  11
Let’s Practice…
3. 3 >
11+𝑘
4
≥ -3
Let’s Practice…
Absolute
Value
Equations &
Inequalities
Review: Absolute Value
• The magnitude of a real number without
regard to its sign.
OR
• Distance of value from zero regardless of sign.
Two Methods to Solving
Absolute Value
• Method One:
– Solve equation like normal
– Consider positive and negative solutions at end
– Warning!!!: Use when there is only a variable is
inside absolute value sign
7 + |𝑥| = 17
Example
5 + 𝑥 = 20
Two Methods to Solving
Absolute Value
• Method Two:
– Remove absolute value
signs by setting one
equation to a negative and
the equation to a positive
– Solve like normal
– Can use ALL the time.
– MUST use when there are
multiple terms inside
absolute value sign
Example
3 + 𝑥 = 15
Solving Tip # 1
After the expression has been rearranged, the
absolute value expression must equal a positive
number.
• If negative = no solution!
Example:
3𝑥 − 6 − 5 = −7
Solving Tip # 2
• Absolute value signs are the same as
parentheses.
• If there is a number directly in front of the
absolute value sign: DISTRIBUTE or simplify
by dividing!
Example:
2 5𝑥 − 4 = 10
Let’s Practice…
Choose your method & solve:
1. 4|v  5| = 16
2. 3|d  4| = 12
Let’s Practice…
Choose your method & solve:
3. |3f + 0.5|  1 = 7
4. 18 = 3 𝑥 + 5
Absolute Value Inequalities
• Absolute Value Inequalities are a form of
Compound Inequalities.
• How to Solve:
– Separate
• Flip inequality sign on
negative side
– Solve
– Graph both solutions.
Example
2𝑥 + 4 ≥ 5
Let’s Practice…
Solve and graph each solution.
1. |f – 1| ≤ 6
2. 2|p + 3| ≥ 10
Equations &
Word Problems
How to:
1. Breathe.
2. Read the whole question.
3. Re-read each sentence and note important
details (numbers!)
4. Draw a picture, if needed.
5. Write an equation.
6. Solve.
Example
Let’s Practice…
At 1pm on Sunny Isles Beach, Juan noticed the
temperature outside was 96 degrees. The
temperature decreased at a steady rate of 4
degrees per hour. Write an equation to model
the situation. At what time was the temperature
80 degrees?
Let’s Practice…
One year, Kent played Play Station 4 for five
fewer hours than Dennis, and Joy played six
hours more than Dennis. Although, the three
friends played on the PS4 for a total of 205 hours.
Write an equation to model the situation. Find the
number of hours each person played on the PS4 in
on year.
Word Problems with Proportions
1. Identify two quantities (usually nouns or
units of measurement).
Examples: cookies, inches, miles, trees, books, etc.
2. Set up fractions….pick which quantity is going
on top and which quantity is going on
bottom.
3. Cross Multiply.
4. Divide to get variable alone.
Example
The windows on a building are proportional to
the size of the building. The height of each
window is 18 in., and the width is 11 in. If the
height of the building is 108 ft, what is the width
of the building?
Calculator = YES
Example
Eric is planning to bake approximately 305
cookies. If 3 pounds of cookie dough make 96
cookies, how many pounds of cookie dough
should he make?
Calculator = YES
Time, Distance and Speed
A cargo plane made a trip to the airshow and
back. The trip there took six hours and the trip
back took four hours. What was the cargo
plane's average speed on the trip there if it
averaged 255 mph on the return trip?
Calculator = YES
Time, Distance and Speed
Eduardo traveled to his friend's house and back.
It took three hours longer to go there than it did
to come back. The average speed on the trip
there was 22 km/h. The average speed on the
way back was 55 km/h. How many hours did the
trip there take?
Calculator = YES
Working Together
Dan can pick forty bushels of apples in eight
hours. Ted can pick the same amount in ten
hours. Find how long it would take them if they
worked together.
Calculator = YES