Solving multi-step inequalities might seem daunting at first, but with a systematic approach and a solid understanding of the underlying principles, you can master this essential algebra skill. This guide will walk you through the process, providing clear explanations, practice problems, and detailed solutions to solidify your understanding.
Understanding the Fundamentals
Before tackling multi-step inequalities, let's refresh our understanding of basic inequality symbols and their meanings:
- <: Less than
- >: Greater than
- ≤: Less than or equal to
- ≥: Greater than or equal to
Remember, the key difference between solving equations and inequalities lies in how we handle multiplication or division by a negative number. When multiplying or dividing both sides of an inequality by a negative number, you must reverse the inequality symbol.
Solving Multi-Step Inequalities: A Step-by-Step Approach
The process of solving multi-step inequalities mirrors that of solving multi-step equations, with the crucial addition of the negative number rule mentioned above. Here's a breakdown of the steps:
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Simplify both sides: Combine like terms on each side of the inequality. This might involve adding, subtracting, multiplying, or dividing terms.
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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: Once the variable term is isolated, perform the necessary operation to solve for the variable. Remember to reverse the inequality symbol if you multiply or divide by a negative number.
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Check your solution: Substitute your solution back into the original inequality to verify that it satisfies the inequality.
Practice Problems with Detailed Solutions
Let's put these steps into practice with some examples:
Problem 1: Solve the inequality 3x + 5 > 11
Solution:
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Simplify: The inequality is already simplified.
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Isolate the variable term: Subtract 5 from both sides: 3x > 6
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Solve for the variable: Divide both sides by 3: x > 2
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Check: Let's try x = 3. 3(3) + 5 = 14, which is greater than 11. This confirms our solution.
Therefore, the solution is x > 2
Problem 2: Solve the inequality -2x + 7 ≤ 1
Solution:
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Simplify: The inequality is already simplified.
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Isolate the variable term: Subtract 7 from both sides: -2x ≤ -6
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Solve for the variable: Divide both sides by -2. Remember to reverse the inequality symbol because we are dividing by a negative number: x ≥ 3
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Check: Let's try x = 3. -2(3) + 7 = 1, which is less than or equal to 1. This confirms our solution.
Therefore, the solution is x ≥ 3
Problem 3: Solve the inequality 4(x - 2) - 3x ≥ 5
Solution:
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Simplify: Distribute the 4: 4x - 8 - 3x ≥ 5. Combine like terms: x - 8 ≥ 5
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Isolate the variable term: Add 8 to both sides: x ≥ 13
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Solve for the variable: The variable is already isolated.
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Check: Let's try x = 14. 4(14 - 2) - 3(14) = 48 - 42 = 6, which is greater than or equal to 5. This confirms our solution.
Therefore, the solution is x ≥ 13
Further Practice and Resources
These examples provide a strong foundation for solving multi-step inequalities. To further enhance your understanding, try creating your own practice problems and working through them step-by-step. Remember to always check your solutions! You can find additional practice problems and resources online through educational websites and textbooks. Consistent practice will build your confidence and mastery of this essential algebraic skill.
