Solving inequalities involves finding the range of values that satisfy the inequality. The goal is similar to solving equations, but with a crucial difference: multiplying or dividing by a negative number reverses the inequality sign.
Here's a breakdown of the steps involved:
Steps to Solve Inequalities
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Simplify both sides of the inequality: Combine like terms and distribute where necessary.
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Isolate the variable term: Use addition or subtraction to get the variable term alone on one side of the inequality. Remember, you can add or subtract the same quantity from both sides without changing the inequality.
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Isolate the variable: Use multiplication or division to get the variable by itself.
- If you multiply or divide by a positive number, the inequality symbol stays the same.
- If you multiply or divide by a negative number, you MUST reverse the inequality symbol.
Rules for Solving Inequalities
The rules for manipulating inequalities are similar to those for equations, with one key exception:
| Operation | Rule | Example |
|---|---|---|
| Addition/Subtraction | You can add or subtract the same quantity from both sides. | If x + 3 > 5, then x + 3 - 3 > 5 - 3, so x > 2 |
| Multiplication/Division (Positive Number) | You can multiply or divide both sides by the same positive quantity. | If 2x < 6, then (2x)/2 < 6/2, so x < 3 |
| Multiplication/Division (Negative Number) | You can multiply or divide both sides by the same negative quantity, BUT you MUST reverse the inequality symbol. | If -2x < 6, then (-2x)/-2 > 6/-2, so x > -3 (Notice the symbol changed from < to >) |
Examples
Example 1: Solve the inequality 3x + 2 < 8
- Subtract 2 from both sides: 3x + 2 - 2 < 8 - 2 => 3x < 6
- Divide both sides by 3: (3x)/3 < 6/3 => x < 2
Therefore, the solution is x < 2.
Example 2: Solve the inequality -2x + 5 ≥ 11
- Subtract 5 from both sides: -2x + 5 - 5 ≥ 11 - 5 => -2x ≥ 6
- Divide both sides by -2 (and reverse the inequality): (-2x)/-2 ≤ 6/-2 => x ≤ -3
Therefore, the solution is x ≤ -3.
Expressing the Solution
Solutions to inequalities can be expressed in a few ways:
- Inequality Notation: x < 2, x ≥ -3
- Interval Notation: (-∞, 2), (-∞, -3] (Parentheses indicate the endpoint is not included; brackets indicate it is.)
- Graphically: A number line with an open circle at 2 pointing to the left for x < 2, or a closed circle at -3 pointing to the left for x ≤ -3.
Solving inequalities requires applying similar algebraic operations as solving equations, but with careful attention to the sign when multiplying or dividing by a negative number. Remember to reverse the inequality sign when multiplying or dividing by a negative value.
