ASVAB Study Guide
Algebra · Lesson 5 of 13

Solving Multiple-Step Inequalities

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

1. Two-Step Inequalities

Using two operations to isolate x

ax + b < c

Using two distinct operations to isolate the variable x.

Example

Solve for x: $$4x - 3 < 13$$

Show solution

Step 1: Add 3 to both sides: $$4x < 16$$.

Step 2: Divide by 4: $$x < 4$$.

Result: x < 4.

Practice problems

  1. Problem 1

    Solve for x: $$2x + 5 > 15$$.

    Show solution

    Answer: x > 5

    Step-by-step multi-step inequality solving:

    • Correct Inequality Result: x > 5

    => Correct Option: B (x > 5).

  2. Problem 2

    Solve for x: $$6x - 8 \le 22$$.

    Show solution

    Answer: $$x \le 5$$

    Step-by-step multi-step inequality solving:

    • Correct Inequality Result: x <= 5

    => Correct Option: A ($$x \le 5$$)

  3. Problem 3

    Solve for x: $$(\frac{x}{3}) + 7 \ge 11$$.

    Show solution

    Answer: $$x \ge 12$$

    Step-by-step multi-step inequality solving:

    • Correct Inequality Result: x >= 12

    => Correct Option: A ($$x \ge 12$$)

Remember

First, add or subtract the constant term 'b'. Next, multiply or divide both sides by the coefficient 'a' to isolate x.

2. With Distributive Prop.

Inequalities with parentheses

$$a(x + b) \ge c$$

Inequalities containing parentheses that require expansion first.

Example

Solve for x: $$5(x + 2) \ge 35$$

Show solution

Step 1: Distribute 5: $$5x + 10 \ge 35$$.

Step 2: Subtract 10: $$5x \ge 25$$.

Step 3: Divide by 5: $$x \ge 5$$.

Result: x >= 5.

Practice problems

  1. Problem 4

    Solve for x: $$3(x - 1) < 12$$.

    Show solution

    Answer: x < 5

    Step-by-step multi-step inequality solving:

    • Correct Inequality Result: x < 5

    => Correct Option: A (x < 5).

  2. Problem 5

    Solve for x: $$2(4x + 3) \le 30$$.

    Show solution

    Answer: $$x \le 3$$

    Step-by-step multi-step inequality solving:

    • Correct Inequality Result: x <= 3

    => Correct Option: B ($$x \le 3$$)

  3. Problem 6

    Solve for x: $$-4(2x - 3) > 20$$.

    Show solution

    Answer: $$x < -1$$

    Step-by-step multi-step inequality solving:

    • Correct Inequality Result: x < -1

    => Correct Option: A ($$x < -1$$)

Remember

Distribute the number outside into the terms inside parentheses first: $$a \times x + a \times b \ge c$$, then solve like a standard two-step inequality.

3. Variables on Both Sides

Moving all x terms to one side

ax + b < cx + d

Inequalities where the variable x appears on both sides of the sign.

Example

Solve for x: $$6x - 4 > 2x + 12$$

Show solution

Step 1: Subtract $$2x$$: $$4x - 4 > 12$$.

Step 2: Add 4: $$4x > 16$$.

Step 3: Divide by 4: $$x > 4$$.

Result: x > 4.

Practice problems

  1. Problem 7

    Solve for x: $$5x + 2 > 3x + 10$$

    Show solution

    Answer: x > 4

    Step-by-step multi-step inequality solving:

    • Correct Inequality Result: x > 4

    => Correct Option: B (x > 4).

  2. Problem 8

    Solve for x: $$8x - 7 \le 3x + 18$$

    Show solution

    Answer: $$x \le 5$$

    Step-by-step multi-step inequality solving:

    • Correct Inequality Result: x <= 5

    => Correct Option: A ($$x \le 5$$)

  3. Problem 9

    Solve for x: $$2x + 15 > 7x - 10$$

    Show solution

    Answer: x < 5

    Step-by-step multi-step inequality solving:

    • Correct Inequality Result: x < 5

    => Correct Option: B (x < 5).

Remember

Group variable terms on one side and constants on the other side using addition/subtraction, then isolate the variable.

4. The Negative Flip

Multi-step with negative coefficients

-ax + b > c

Multi-step inequalities where the coefficient of x is negative.

Example

Solve for x: $$-3x + 8 > 23$$

Show solution

Step 1: Subtract 8: $$-3x > 15$$.

Step 2: Divide by -3 and FLIP sign: x < -5.

Result: x < -5.

Practice problems

  1. Problem 10

    Solve for x: $$-4x + 6 < 22$$.

    Show solution

    Answer: $$x > -4$$

    Step-by-step multi-step inequality solving:

    • Correct Inequality Result: x > -4

    => Correct Option: B ($$x > -4$$)

  2. Problem 11

    Solve for x: $$3 - 6x \ge 33$$.

    Show solution

    Answer: $$x \le -5$$

    Step-by-step multi-step inequality solving:

    • Correct Inequality Result: x <= -5

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

  3. Problem 12

    Solve for x: $$-3(x + 2) + 5x \le 8$$.

    Show solution

    Answer: $$x \le 7$$

    Step-by-step multi-step inequality solving:

    • Correct Inequality Result: x <= 7

    => Correct Option: A ($$x \le 7$$)

Remember

Crucial: If the final step involves multiplying or dividing by a negative number, always REVERSE (flip) the inequality sign.