Solving One-Step Inequalities - Sixth Grade Mathematics

Solving one-step inequalities applies inverse operations to isolate the variable, following equation-solving properties while applying the critical sign reversal rule when multiplying or dividing by negative numbers.

Addition and Subtraction Properties of Inequality

Adding or subtracting the exact same number from both sides of an inequality preserves the truth of the inequality without altering the direction of the inequality symbol.

Addition Property of Inequality:
If a < b, then a + c < b + c.

Subtraction Property of Inequality:
If a < b, then a - c < b - c.

Worked Examples:
Example A: Solve  x + 6 > 11
Subtract 6 from both sides:
x + 6 - 6 > 11 - 6
x > 5.
Graph: Open circle at 5, shade to the right.

Example B: Solve  y - 4 <= 9
Add 4 to both sides:
y - 4 + 4 <= 9 + 4
y <= 13.
Graph: Closed circle at 13, shade to the left.

Multiplying and Dividing by Positive Numbers

When multiplying or dividing both sides of an inequality by a POSITIVE number, the inequality symbol remains unchanged.

Example C: Solve  4x >= 28
Divide both sides by positive 4:
4x / 4 >= 28 / 4
x >= 7.
Graph: Closed circle at 7, shade to the right.

Example D: Solve  m / 3 < 5
Multiply both sides by positive 3:
(m / 3) * 3 < 5 * 3
m < 15.
Graph: Open circle at 15, shade to the left.

The Negative Multiplication/Division Reversal Rule

The single most critical rule in all of inequality algebra states: Whenever you MULTIPLY or DIVIDE both sides of an inequality by a NEGATIVE number, you MUST REVERSE (FLIP) the direction of the inequality symbol!

+-------------------------------------------------------------------------+
|                  THE NEGATIVE REVERSAL THEOREM                          |
|                                                                         |
|   If  a < b  and  c is NEGATIVE (c < 0),                                |
|   then  a * c  >  b * c   and   a / c  >  b / c.                        |
|                                                                         |
|   Symbol Flips:                                                         |
|   <  becomes  >              <=  becomes  >=                            |
|   >  becomes  <              >=  becomes  <=                            |
+-------------------------------------------------------------------------+

Mathematical Proof of the Sign Reversal

Consider the true numerical inequality: 2 < 5. Multiply both sides by -1: Left side: 2 * (-1) = -2. Right side: 5 * (-1) = -5. Now compare -2 and -5 on a number line: -2 is to the RIGHT of -5! Therefore, -2 > -5. If you did not flip the symbol, you would state -2 < -5, which is mathematically false. Multiplying by a negative reverses directional positions along the number line, requiring the inequality symbol to flip.

Worked Example: Solve  -3x <= 15
Step 1: Divide both sides by -3.
Step 2: REVERSE the inequality symbol from <= to >=!
        (-3x) / (-3)  >=  15 / (-3)
        x             >=  -5.
Check: Test x = 0 (which is >= -5): -3(0) <= 15 --> 0 <= 15. (True!)

Exercises

Exercise 1. Solve each inequality and state the solution set: (a) x + 9 < 15; (b) y - 7 >= 12; (c) 5m <= 35; (d) w / 4 > 6.

Exercise 2. Solve the inequality requiring sign reversal: -4k < 24. Show the symbol flip explicitly.

Exercise 3. A student solves -2x >= 10 and writes the answer as x >= -5. Explain the error, provide the correct solution, and test a value to prove why the student's answer was incorrect.

Solutions

Solution to Exercise 1. For (a) x + 9 < 15: subtract 9 from both sides: x < 6. For (b) y - 7 >= 12: add 7 to both sides: y >= 19. For (c) 5m <= 35: divide both sides by positive 5 (no flip): m <= 7. For (d) w / 4 > 6: multiply both sides by positive 4 (no flip): w > 24.

Solution to Exercise 2. For -4k < 24: divide both sides by -4. Because you are dividing by a negative number, the less-than symbol (<) flips to greater-than (>): k > 24 / (-4), which simplifies to k > -6.

Solution to Exercise 3. The student forgot to reverse the inequality symbol when dividing by negative 2. In -2x >= 10, dividing both sides by -2 must flip the symbol from >= to <=, producing the correct solution x <= -5. To prove the student's answer (x >= -5) is false, test a number that satisfies the student's claim but violates the true inequality, such as x = 0 (since 0 >= -5). Substitute x = 0 into the original inequality: -2(0) >= 10, which gives 0 >= 10. Since 0 is NOT greater than or equal to 10, the student's solution set includes false values. Testing x = -6 in the correct solution (x <= -5): -2(-6) = 12 >= 10, which is True!