Solving One-Step Inequalities
Solving one-step inequalities is a foundational Grade 9 Algebra 1 skill in California Reveal Math (Unit 5: Linear Inequalities). Like equations, you isolate the variable using inverse operations. The critical rule: when you multiply or divide both sides by a negative number, you must flip the inequality symbol. For example, dividing -2y <= 8 by -2 gives y >= -4 because the direction reverses. Adding or subtracting never changes the sign, only multiplication or division by negatives triggers the flip.
Key Concepts
Property Solving one step inequalities is almost identical to solving one step equations: you use inverse operations to isolate the variable. Addition/Subtraction: You can add or subtract the same number from both sides. The inequality sign does not change . Multiplication/Division by a Positive: You can multiply or divide both sides by a positive number. The inequality sign does not change . Multiplication/Division by a Negative: If you multiply or divide both sides by a negative number, you MUST reverse the inequality symbol ($<$ becomes $ $, $\geq$ becomes $\leq$).
Examples Using Subtraction: Solve $x + 5 \geq 12$. Subtract 5 from both sides: $x \geq 7$. (Sign stays the same) Dividing by a Positive: Solve $3x 15$. Divide both sides by 3: $x 5$. (Sign stays the same) Dividing by a Negative: Solve $ 2y \leq 8$. Divide both sides by 2. Because you divided by a negative, reverse the sign: $y \geq 4$.
Explanation For the most part, you can treat an inequality symbol just like an equal sign when trying to get a variable by itself. The only major trap is the "Negative Rule." Why must we flip the sign? Think about the number line: $4 2$ is true. But if we multiply both sides by 1, we get $ 4$ and $ 2$. Since 4 is further to the left on the number line, it is now smaller, so we must flip the sign to make the statement true: $ 4 < 2$.
Common Questions
When do you flip the inequality sign?
Only when multiplying or dividing both sides by a negative number. For example, dividing -2y <= 8 by -2 flips the sign: y >= -4. Adding or subtracting from both sides never changes the inequality direction.
How do you solve x + 5 >= 12?
Subtract 5 from both sides: x >= 7. The inequality sign stays the same because you are subtracting, not multiplying or dividing by a negative.
How do you solve -2y <= 8?
Divide both sides by -2. Because you are dividing by a negative, reverse the sign: y >= -4. The sign flips from <= to >=.
Why does dividing by a negative number flip the inequality?
Consider 4 > 2. Multiply both sides by -1 to get -4 and -2. Since -4 is further left on the number line, -4 < -2. The relationship reversed, so the symbol must flip.
How do you solve 3x > 15?
Divide both sides by 3 (a positive number). The sign stays the same: x > 5. No flip needed because the divisor is positive.