Solving multi-step inequalities might seem daunting at first, but with a systematic approach and a solid understanding of the underlying principles, you'll master this essential algebra skill in no time. This guide provides a step-by-step walkthrough, practical examples, and a worksheet with answers to solidify your understanding.
Understanding the Fundamentals
Before tackling multi-step inequalities, let's refresh our understanding of basic inequality symbols:
- <: Less than
- >: Greater than
- ≤: Less than or equal to
- ≥: Greater than or equal to
Remember, inequalities represent a range of values, unlike equations which have a single solution. When solving inequalities, the goal is to isolate the variable to determine the range of values that satisfy the inequality.
Solving Multi-Step Inequalities: A Step-by-Step Approach
Solving multi-step inequalities involves applying the same principles as solving multi-step equations, with one crucial difference: When multiplying or dividing by a negative number, you must reverse the inequality sign.
Here's a breakdown of the process:
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Simplify both sides: Combine like terms on each side of the inequality.
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Isolate the variable term: Use inverse operations (addition, subtraction, multiplication, division) to move all terms containing the variable to one side of the inequality and all constant terms to the other side. Remember to perform the same operation on both sides.
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Solve for the variable: Divide or multiply both sides by the coefficient of the variable. Crucially, reverse the inequality sign if you multiply or divide by a negative number.
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Graph the solution (optional): Represent the solution set on a number line. Use an open circle (○) for < or > and a closed circle (●) for ≤ or ≥.
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Check your solution (recommended): Choose a value within your solution set and substitute it back into the original inequality to verify it's a valid solution.
Example Problems
Let's work through a couple of examples to illustrate the process:
Example 1:
Solve the inequality: 3x - 5 > 7
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Add 5 to both sides: 3x > 12
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Divide both sides by 3: x > 4
Solution: x > 4. This means any value greater than 4 satisfies the inequality.
Example 2:
Solve the inequality: -2x + 4 ≤ 10
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Subtract 4 from both sides: -2x ≤ 6
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Divide both sides by -2 and reverse the inequality sign: x ≥ -3
Solution: x ≥ -3. This means any value greater than or equal to -3 satisfies the inequality.
Example 3 (Involving Distribution):
Solve: 2(x + 3) - 5 < 9
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Distribute: 2x + 6 - 5 < 9
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Simplify: 2x + 1 < 9
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Subtract 1 from both sides: 2x < 8
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Divide by 2: x < 4
Worksheet: Solving Multi-Step Inequalities
Now it's your turn! Try solving these multi-step inequalities. Remember to show your work. Answers are provided below.
- 5x + 10 > 25
- -3x - 6 ≤ 9
- 4(x - 2) ≥ 12
- -2(x + 1) + 5 < 7
- 3x - 7 > 2x + 4
Answers to Worksheet
- x > 3
- x ≥ -5
- x ≥ 5
- x > -2
- x > 11
This guide and worksheet provide a strong foundation for understanding and solving multi-step inequalities. Practice is key! The more you work through these types of problems, the more comfortable and confident you'll become. Remember to always check your answers. If you're still struggling, consider seeking additional help from a teacher, tutor, or online resources.
