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Equations & Inequalities

Standards: 6.EE.B.5–8 (Solving Equations, Writing Equations, Inequalities)

Directions: Read the vocabulary box and worked examples. Then solve each equation or inequality. Remember: do the same thing to both sides. Show your steps.

Vocabulary Helper

⚖️ Equation
Two expressions that are equal, joined by an = sign, like x + 4 = 10.
✅ Solution
The value of the variable that makes the equation true.
🔤 Variable
A letter that stands for an unknown number, like x or n.
↩️ Inverse Operation
The opposite operation used to undo. Add undoes subtract; multiply undoes divide.
↔️ Inequality
Compares with >, <, ≥, or ≤ instead of =.
🔢 Coefficient
The number multiplied by the variable. In 3x, the coefficient is 3.

Worked Examples

EXAMPLE A — One-Step Equation

Solve x + 4 = 10:

  1. The inverse of + 4 is − 4.
  2. Subtract 4 from both sides: x + 4 − 4 = 10 − 4
  3. x = 6. Check: 6 + 4 = 10 ✓
EXAMPLE B — Inequality on a Number Line

Graph x > 3:

  1. Put an open circle on 3 (because it is >, not ≥).
  2. Shade to the right (all numbers bigger than 3).

Sentence Frames

To solve x − 5 = 12, I use the inverse operation and 5 to both sides, so x = .
In the equation 4x = 20, the coefficient is , so I both sides by 4.
The inequality x ≤ 7 uses a circle because 7 a solution.

Guided Practice

1. Solve: x + 6 = 15

Hint: Subtract 6 from both sides.

2. Solve: y − 8 = 3

Hint: Add 8 to both sides.

3. Solve: 4x = 20

Hint: Divide both sides by 4.

4. Solve: n ÷ 3 = 6 (or n/3 = 6)

Hint: Multiply both sides by 3.

5. Is x = 4 a solution to x + 5 = 9? (yes/no)

Hint: Substitute 4 for x and check if both sides match.

6. Solve: 3x + 7 = 28

Hint: Subtract 7 first, then divide by 3.

7. Write an equation: "A number plus 5 equals 12." Then solve.

Hint: x + 5 = 12.

8. Solve the inequality: x + 2 > 5

Hint: Subtract 2 from both sides.

9. Solve the inequality: 2x ≤ 10

Hint: Divide both sides by 2.

10. Graph the solution to x < 4 on a number line. (Open or closed circle?)

Hint: < uses an open circle; shade left.

Answer Key

  1. x = 9
  2. y = 11
  3. x = 5
  4. n = 18
  5. Yes — 4 + 5 = 9 ✓
  6. x = 7 — 3x = 21, x = 7
  7. x + 5 = 12, x = 7
  8. x > 3
  9. x ≤ 5
  10. x < 4 — open circle on 4, shade left