ASVAB Study Guide
Algebra · Lesson 4 of 13

Solving One-Step Inequalities

Key rules and formulas, worked examples and 12 practice problems with step-by-step solutions.

1. Addition / Subtraction

Inequalities involving adding or subtracting

$$x + a > b$$ or $$x - a < b$$

Inequalities involving adding or subtracting a constant.

Example

Solve for x: $$x - 5 > 12$$

Show solution

Step 1: Add 5 to both sides.

$$x > 12 + 5 \Rightarrow x > 17$$.

Note: Inequality sign remains unchanged.

Result: x > 17.

Practice problems

  1. Problem 1

    Solve for x: $$x + 7 < 15$$.

    Show solution

    Answer: x < 8

    Step-by-step inequality solving:

    • Correct Inequality Result: x < 8

    => Correct Option: B (x < 8).

  2. Problem 2

    Solve for x: $$x - 12 \ge -4$$.

    Show solution

    Answer: $$x \ge 8$$

    Step-by-step inequality solving:

    • Correct Inequality Result: x >= 8

    => Correct Option: B ($$x \ge 8$$)

  3. Problem 3

    Solve for x: $$5 + x \le -9$$.

    Show solution

    Answer: $$x \le -14$$

    Step-by-step inequality solving:

    • Correct Inequality Result: x <= -14

    => Correct Option: B ($$x \le -14$$)

Remember

Perform the inverse operation (Add or Subtract) on both sides to isolate x.

Rule: Adding or subtracting ANY number NEVER changes the inequality sign direction.

2. Multiplication (Pos)

Multiplying by a positive number

$$a \times x > b$$ (where $$a > 0$$)

Multiplying or dividing by a positive number to isolate x.

Example

Solve for x: $$4x \ge 20$$

Show solution

Step 1: Divide both sides by positive 4.

$$x \ge \frac{20}{4} \Rightarrow x \ge 5$$.

Note: Sign direction stays the same.

Result: x >= 5.

Practice problems

  1. Problem 4

    Solve for x: $$3x < 18$$.

    Show solution

    Answer: x < 6

    Step-by-step inequality solving:

    • Correct Inequality Result: x < 6

    => Correct Option: B (x < 6).

  2. Problem 5

    Solve for x: $$6x \ge -42$$.

    Show solution

    Answer: $$x \ge -7$$

    Step-by-step inequality solving:

    • Correct Inequality Result: x >= -7

    => Correct Option: B ($$x \ge -7$$)

  3. Problem 6

    Solve for x: $$0.5x > 9$$.

    Show solution

    Answer: x > 18

    Step-by-step inequality solving:

    • Correct Inequality Result: x > 18

    => Correct Option: B (x > 18).

Remember

Divide both sides by the positive coefficient 'a'.

Rule: When multiplying or dividing by a POSITIVE number, the inequality sign stays the SAME.

3. Division (Pos)

Dividing by a positive number

x / a < b (where a > 0)

Dividing by a positive constant (x is in the numerator).

Example

Solve for x: $$\frac{x}{3} < 7$$

Show solution

Step 1: Multiply both sides by positive 3.

$$x < 7 \times 3 \Rightarrow x < 21$$.

Note: Sign direction stays the same.

Result: x < 21.

Practice problems

  1. Problem 7

    Solve for x: $$\frac{x}{5} > 4$$.

    Show solution

    Answer: x > 20

    Step-by-step inequality solving:

    • Correct Inequality Result: x > 20

    => Correct Option: B (x > 20).

  2. Problem 8

    Solve for x: $$\frac{x}{2} \le -8$$.

    Show solution

    Answer: $$x \le -16$$

    Step-by-step inequality solving:

    • Correct Inequality Result: x <= -16

    => Correct Option: B ($$x \le -16$$)

  3. Problem 9

    Solve for x: $$\frac{x}{0.4} > -5$$.

    Show solution

    Answer: $$x > -2$$

    Step-by-step inequality solving:

    • Correct Inequality Result: x > -2

    => Correct Option: A ($$x > -2$$)

Remember

Multiply both sides by the positive denominator 'a' to clear the fraction.

Rule: Since the multiplier is positive, the inequality sign stays the SAME.

4. Negative Rule

Multiplying or Dividing by negatives

$$-a \times x < b$$ or $$\frac{x}{(-a)} > b$$ (where $$-a < 0$$)

Multiplying or dividing both sides by a negative number.

Example

Solve for x: $$-3x < 15$$

Show solution

Step 1: Divide both sides by -3.

Step 2: FLIP the inequality sign from '<' to '>'.

$$x > \frac{15}{(-3)} \Rightarrow x > -5$$.

Result: x > -5.

Practice problems

  1. Problem 10

    Solve for x: $$-2x > 10$$.

    Show solution

    Answer: $$x < -5$$

    Step-by-step inequality solving:

    • Correct Inequality Result: x < -5

    => Correct Option: B ($$x < -5$$)

  2. Problem 11

    Solve for x: $$-5x \le -35$$.

    Show solution

    Answer: $$x \ge 7$$

    Step-by-step inequality solving:

    • Correct Inequality Result: x >= 7

    => Correct Option: B ($$x \ge 7$$)

  3. Problem 12

    Solve for x: $$\frac{x}{-4} \ge 6$$.

    Show solution

    Answer: $$x \le -24$$

    Step-by-step inequality solving:

    • Correct Inequality Result: x <= -24

    => Correct Option: A ($$x \le -24$$)

Remember

CRUCIAL RULE: Whenever you multiply or divide both sides of an inequality by a NEGATIVE number, you MUST FLIP (reverse) the inequality sign:

  • '>' becomes '<'
  • '<' becomes '>'
  • '>=' becomes '<='
  • '<=' becomes '>='