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Equations & Inequalities

Interactive study guide — Reveal Math Unit 8 · Standard 6.EE

8.1 — Understand Equations & Solutions

An equation is a math sentence with an equals sign. A solution is a value that makes the equation true.

Key Idea

To check if a value is a solution, substitute it into the equation and see if both sides are equal.

Example: Is x = 4 a solution to x + 3 = 7?
Substitute: 4 + 3 = 7 ✓ Yes!
Example: Is x = 5 a solution to 2x = 12?
Substitute: 2(5) = 10 ≠ 12. No!

Practice

Is x = 6 a solution to x + 2 = 8?

No, because 6 + 2 = 10
Yes, because 6 + 2 = 8
No, because 6 − 2 = 4

Which value is a solution to 3x = 15?

x = 3
x = 12
x = 5

Is x = 7 a solution to x − 3 = 4?

Yes, because 7 − 3 = 4
No, because 7 − 3 = 5
Yes, because 7 + 3 = 10
Score
0/3

8.2 — Solve with Addition & Subtraction

Write equations from word problems and solve using inverse operations.

Key Idea

To solve, do the opposite operation to both sides to isolate the variable.

Example: x + 5 = 12
Subtract 5 from both sides: x = 12 − 5 = 7
Example: x − 7 = 9
Add 7 to both sides: x = 9 + 7 = 16
Word Problem: "A number increased by 5 is 12" → x + 5 = 12 → x = 7

Practice

Solve: x + 9 = 14

x = 23
x = 5
x = 9

Solve: x − 8 = 3

x = 11
x = 5
x = −5

"A number decreased by 4 is 10." Which equation matches?

x + 4 = 10
4x = 10
x − 4 = 10
Score
0/3

8.3 — Solve with Multiplication & Division

Solve equations using inverse operations — multiply to undo division, divide to undo multiplication.

Key Idea

Use the inverse operation to get the variable alone.

Example: 4x = 20
Divide both sides by 4: x = 20 ÷ 4 = 5
Example: x ÷ 6 = 3
Multiply both sides by 6: x = 3 × 6 = 18
Word Problem: "Four times a number is 20" → 4x = 20 → x = 5

Practice

Solve: 6x = 42

x = 48
x = 36
x = 7

Solve: x ÷ 5 = 8

x = 40
x = 3
x = 13

"A number divided by 3 is 9." What is the number?

x = 3
x = 27
x = 12
Score
0/3

8.4 — Write & Represent Inequalities

Inequalities use >, <, ≥, ≤ to compare values. They can be shown on a number line.

Key Idea

An inequality says one side is greater than or less than the other. Use an open circle for > or < and a closed circle for ≥ or ≤.

Example: x > 3 means all numbers greater than 3 (open circle at 3, arrow right)
Example: x ≤ 5 means 5 and all numbers less than 5 (closed circle at 5, arrow left)
Real World: "You must be at least 48 inches tall" → h ≥ 48

Practice

"A number is greater than 7." Which inequality matches?

x ≤ 7
x > 7
x = 7

For x ≥ 4, which circle goes on the number line at 4?

Closed (filled) circle
Open (empty) circle
No circle needed

"You can spend at most $20." Which inequality fits?

x > 20
x ≥ 20
x ≤ 20
Score
0/3

8.5 — Solve Inequalities

Inequalities have many solutions, not just one. Solve and check using reasoning.

Key Idea

Solve inequalities the same way you solve equations — but the answer is a range of values, not just one number.

Example: x + 2 > 5
Subtract 2: x > 3
Solutions: 4, 5, 6, 3.5, 100 … any number greater than 3!
Example: 3x ≤ 12
Divide by 3: x ≤ 4
Solutions: 4, 3, 2, 0, −1 … any number that is 4 or less!

Practice

Solve: x + 4 > 10

x > 6
x > 14
x < 6

Solve: 2x ≤ 16

x ≤ 32
x ≥ 8
x ≤ 8

Which is a solution to x − 3 ≥ 5?

x = 7
x = 8
x = 2
Score
0/3

Vocabulary

Tap a card to reveal the definition.

Equation
A math sentence with an equals sign showing two expressions are equal.
Solution
A value that makes an equation or inequality true.
Variable
A letter that represents an unknown number.
Inverse Operation
The opposite operation used to undo another (+ ↔ −, × ↔ ÷).
Inequality
A math sentence using >, <, ≥, or ≤ to compare two expressions.
Coefficient
The number multiplied by a variable (the 4 in 4x).
Expression
A combination of numbers, variables, and operations (no equals sign).
Open Circle
Used on a number line for > or < (value is NOT included).
Closed Circle
Used on a number line for ≥ or ≤ (value IS included).