Equation Inequality Notes Handout
Algebra Foundations
Solving One-Step Equations
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The Core Goal: Isolation
To solve an equation, we must ISOLATE the variable (get it completely by itself). We undo operations using inverse operations.
Golden Rule: Whatever you do to one side of the equal sign, you MUST do to the other side!
Inverse (Opposite) Operations
Addition (+) ↔ Subtr. (−)
Mult. (×) ↔ Div. (÷)
Mr. Davis’s Tips to Help Solve Equations:
Essential Strategy
-
Drop your lines around your equal sign (keeps sides balanced)
-
Box in the variable you are solving for (with sign & coefficient)
-
Draw a horizontal line after each step so you don't get confused
-
WRITE NEATLY!!!
Model Demonstration
x + 8 = 18
Box variable & drop lines!
Guided Practice Apply all 4 of Mr. Davis's steps
- x - 8 = 15
x =
- 4m = 28
m =
- \( \frac{k}{6} = 7 \)
k =
Independent Practice (6 Problems) Show all work neatly
- p + 14 = 30
p = _____
- y - 9 = 17
y = _____
- 5w = 45
w = _____
- \( \frac{a}{8} = 4 \)
a = _____
- -6b = 42
b = _____
- \( \frac{c}{-3} = 7 \)
c = _____
Check Your Work: Substitute your answer back into the original equation to verify both sides balance!
Algebra Foundations
Solving & Graphing Inequalities
Back: Inequalities
Equations vs. Inequalities: Core Difference
An equation has exactly ONE specific solution. An inequality has an INFINITE set of solutions! Because there are many solutions, we visually represent them by graphing on a number line.
Graphing Guide: Open Circle vs. Closed Dot
OPEN CIRCLE Not Included
Symbols: < (Less Than) or > (Greater Than)
The boundary number is NOT part of the solution set.
x > 3 :
◂
3
▸
CLOSED DOT (Solid) Included
Symbols: ≤ (Less Than / Equal) or ≥ (Greater / Equal)
The boundary number IS part of the solution set.
x ≤ 3 :
◂
3
▸
The Golden Inequality Rule (FLIP ALERT!):
When you MULTIPLY or DIVIDE both sides by a NEGATIVE NUMBER, you MUST FLIP the inequality sign! (> flips to < | ≤ flips to ≥)
Guided Practice (2 Problems)
Solve & graph together
- x - 4 < 7
x __________
←
9
10
11
12
13
→
- -2k ≥ 10
k __________
←
-7
-6
-5
-4
-3
→
Independent Practice (4 Problems)
Solve, determine circle, and graph
- p + 6 ≥ 10
p __________
←
2
3
4
5
6
→
- y - 3 < 5
y __________
←
6
7
8
9
10
→
- 4m ≤ 20
m __________
←
3
4
5
6
7
→
- \( \frac{z}{-2} > 3 \)
z __________
←
-8
-7
-6
-5
-4
→
Pro-Tip for Shading: If the variable is on the left side (e.g. \( x > 3 \)), the arrow always points in the same direction as the inequality symbol!
