A multi-step inequality is an inequality that requires more than one step to solve for the unknown variable.
Understanding Multi-Step Inequalities
Unlike simple inequalities that can be solved in a single step (like adding a constant to both sides), multi-step inequalities involve a combination of operations, such as:
- Distribution: Multiplying a number by a group of terms inside parentheses.
- Combining Like Terms: Simplifying each side of the inequality by adding or subtracting terms with the same variable or constant.
- Adding or Subtracting: Adding or subtracting the same number from both sides of the inequality to isolate the variable term.
- Multiplying or Dividing: Multiplying or dividing both sides of the inequality by the same number to solve for the variable. Important Note: Multiplying or dividing by a negative number requires flipping the inequality sign.
Example of a Multi-Step Inequality
Consider the following inequality:
2(x + 3) - 5 > 7
To solve this, you would need to:
- Distribute:
2x + 6 - 5 > 7 - Combine Like Terms:
2x + 1 > 7 - Subtract:
2x > 6 - Divide:
x > 3
As you can see, solving this inequality requires multiple steps, making it a multi-step inequality.
Key Differences from Equations
The primary difference between solving multi-step inequalities and multi-step equations lies in how multiplication or division by a negative number affects the result. In an equation, the sign remains the same. However, in an inequality, when you multiply or divide by a negative number, you must reverse the inequality sign.
For example:
-2x < 6
To solve for x, you would divide both sides by -2. Since you're dividing by a negative number, you must flip the inequality sign:
x > -3
Solving Multi-Step Inequalities: A Step-by-Step Guide
- Simplify both sides: Distribute and combine like terms on each side of the inequality.
- Isolate the variable term: Use addition or subtraction to get the variable term alone on one side of the inequality.
- Solve for the variable: Use multiplication or division to solve for the variable. Remember to flip the inequality sign if you multiply or divide by a negative number.
- Graph the Solution (Optional): Represent the solution on a number line.
Common Mistakes to Avoid
- Forgetting to distribute correctly: Ensure you multiply the number outside the parentheses by every term inside.
- Not flipping the inequality sign: Remember to reverse the sign when multiplying or dividing by a negative number.
- Incorrectly combining like terms: Double-check that you're only combining terms with the same variable or constant.
