Topic 2: Solving Equations
& Inequalities in One
Variable
Algebra 1

Table of Contents
1. Solving Multi-Step Equations Advanced
2. Special Types of Equations
3. Literal Equations
4. Graphing Inequalities
5. Solving Inequalities
6. Solving Compound Inequalities
7. Absolute Value Equations & Inequalities
8. Equations & Word Problems

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
3
(4x – 8) = 16

Example
6(4 + 3x) – 10(x – 5) = 30 – 2(x + 4)

Let’s Practice…
1. 4(5 + 2x) + 7x = -5(x + 7)

2.
Let’s Practice…
2
3 𝑥 + 6 = −
5
2
(4𝑥 − 2)

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
2
( 10y + 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
2
(10 𝑥 + 8)

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!

Solve:
1. ax – b =c for x
Examples
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

What is an Inequality?!
Inequality: statement that two quantities are not equal.
Symbols of comparison:
< > ≤ ≥ ≠

t > 4
Graphing Inequalities
Remember:
- If you take the time to draw the line, you take the time to fill it in!
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
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 4. 𝑥
3
< 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.

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
≥ -3
4

Let’s Practice…

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

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