Two-Step Inequalities Worksheet Answers Key: Simplified Solutions
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Two-step inequalities can often be a challenging area of mathematics for many students. The concept, however, is critical as it lays the foundation for understanding more complex algebraic expressions and inequalities. In this article, we will simplify two-step inequalities, explore the answers key for worksheets, and provide a clear explanation of the steps involved in solving them. Let's dive in! 📚✨
What is a Two-Step Inequality?
A two-step inequality is an algebraic expression that requires two operations to isolate the variable. Similar to equations, inequalities are statements that one side is larger (or smaller) than the other. They use symbols such as:
- ( < ): less than
- ( > ): greater than
- ( \leq ): less than or equal to
- ( \geq ): greater than or equal to
Example of a Two-Step Inequality
For instance, consider the inequality:
[ 3x + 5 < 14 ]
In this inequality, we need to perform two steps to solve for ( x ).
Steps to Solve Two-Step Inequalities
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Subtract or Add: First, isolate the term that contains the variable. If the inequality includes a constant term on the same side as the variable, you need to move it first.
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Multiply or Divide: Next, perform the operation that involves the variable. Be cautious—if you multiply or divide both sides by a negative number, the direction of the inequality sign changes!
Solving the Example
Let’s solve the example inequality ( 3x + 5 < 14 ) step by step:
Step 1: Subtract 5 from both sides
[ 3x < 14 - 5 ]
[ 3x < 9 ]
Step 2: Divide both sides by 3
[ x < \frac{9}{3} ]
[ x < 3 ]
This means that the solution to the inequality is any number less than 3. 🎉
Important Note
"When solving inequalities, always remember to reverse the inequality sign if you multiply or divide by a negative number."
Example Problems and Solutions
Now, let’s look at a couple of example problems with solutions to reinforce your understanding of two-step inequalities.
Example 1
Solve the inequality:
[ 4x - 7 \geq 9 ]
Solution
Step 1: Add 7 to both sides
[ 4x \geq 9 + 7 ]
[ 4x \geq 16 ]
Step 2: Divide both sides by 4
[ x \geq 4 ]
Example 2
Solve the inequality:
[ 2x + 6 < 14 ]
Solution
Step 1: Subtract 6 from both sides
[ 2x < 14 - 6 ]
[ 2x < 8 ]
Step 2: Divide both sides by 2
[ x < 4 ]
Two-Step Inequalities Worksheet Answers Key
To assist in learning, here’s a concise table showcasing common two-step inequalities along with their solutions. This can serve as an answer key for worksheets.
<table> <tr> <th>Problem</th> <th>Solution</th> </tr> <tr> <td>2x + 3 < 11</td> <td>x < 4</td> </tr> <tr> <td>5x - 2 > 8</td> <td>x > 2</td> </tr> <tr> <td>-3x + 4 ≤ 10</td> <td>x ≥ -2</td> </tr> <tr> <td>6x - 1 ≥ 5</td> <td>x ≥ 1</td> </tr> </table>
Practice Makes Perfect! 📝
To master two-step inequalities, practice is key. Here are some additional inequalities for practice:
- ( 7x + 5 < 26 )
- ( 4x - 8 > 0 )
- ( -2x + 3 ≤ 1 )
- ( 5 - 3x ≥ 14 )
Conclusion
Understanding two-step inequalities is essential for mastering algebra. With practice, the process of isolating the variable becomes more manageable and intuitive. As you work through different problems, keep the key concepts in mind, such as the need to reverse the inequality sign when multiplying or dividing by negative numbers. The answers key provided in this article will serve as a guide to check your work and ensure accuracy.
Whether you're preparing for an exam or simply looking to sharpen your math skills, remember that consistency and practice are the best strategies for success. Happy learning! 🌟