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Solve Inequalities With Brackets: A Step-by-Step Guide

7 min read 11-15- 2024
Solve Inequalities With Brackets: A Step-by-Step Guide

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Solving inequalities can be a challenging task for many students, especially when brackets are involved. This guide will help you navigate the process, breaking it down into easy-to-understand steps. We’ll cover everything from the basics of inequalities to more complex scenarios involving brackets. 🧮 Let’s dive in!

Understanding Inequalities

Inequalities are mathematical expressions that indicate the relationship between two values when they are not equal. The most common inequality symbols are:

  • < (less than)
  • > (greater than)
  • ≤ (less than or equal to)
  • ≥ (greater than or equal to)

Basic Example

For instance, if we have the inequality:

x + 5 > 10

We can solve it by isolating ( x ):

x > 5

This means that any value of ( x ) greater than 5 will satisfy this inequality. ✅

What Are Brackets?

Brackets, or parentheses, indicate that operations inside them should be performed first. For example:

2(x + 3) > 12

In this inequality, we need to handle the expression in brackets before moving forward.

Types of Brackets

  1. Round Brackets ( )
  2. Square Brackets [ ]
  3. Curly Brackets { }

Each type of bracket has its specific use, but for the purpose of inequalities, round brackets are the most common.

Solving Inequalities Involving Brackets

Now, let’s go step-by-step through solving inequalities with brackets. Here’s how to do it:

Step 1: Distribute the Brackets

When you encounter an expression with brackets, the first thing you should do is distribute (or expand) the brackets.

Example:

2(x + 3) > 12

Distributing the 2:

2x + 6 > 12

Step 2: Isolate the Variable

Next, isolate the variable on one side of the inequality. To do this, perform the same operations on both sides.

Continuing our example:

2x + 6 > 12

Subtract 6 from both sides:

2x > 6

Step 3: Divide by the Coefficient

Now, divide by the coefficient of the variable (in this case, 2):

x > 3

Step 4: Check Your Solution

It’s crucial to check your solution by plugging it back into the original inequality to ensure it holds true.

For our solution ( x > 3 ):

If ( x = 4 ):

2(4 + 3) > 12
2(7) > 12
14 > 12 ✅

This means our solution is correct!

Handling Complex Inequalities

Example 1: Compound Inequalities

Let’s consider a slightly more complex example:

3(x - 1) < 2(x + 4)

Step 1: Distribute the Brackets

Distributing gives us:

3x - 3 < 2x + 8

Step 2: Isolate the Variable

Subtract ( 2x ) from both sides:

3x - 2x - 3 < 8

This simplifies to:

x - 3 < 8

Add 3 to both sides:

x < 11

Example 2: Inequalities with Square Brackets

Square brackets often indicate a closed interval, meaning the endpoint is included in the solution. For instance, if we have:

[x - 2] ≥ 0

This means we need to consider when ( x - 2 = 0 ):

x ≥ 2

Important Note:

In cases where the square bracket is included, always remember to check whether the endpoint is part of the solution.

Summary Table of Steps

To provide a clear overview, here’s a summarized table of the steps to follow when solving inequalities with brackets:

<table> <tr> <th>Step</th> <th>Action</th> </tr> <tr> <td>1</td> <td>Distribute the brackets.</td> </tr> <tr> <td>2</td> <td>Isolate the variable.</td> </tr> <tr> <td>3</td> <td>Divide by the coefficient.</td> </tr> <tr> <td>4</td> <td>Check your solution.</td> </tr> </table>

Practice Problems

To master these concepts, practice is essential. Here are some inequalities with brackets for you to solve:

  1. 4(2x - 1) > 8
  2. 5(x + 3) ≤ 2(3x + 1)
  3. 3(x - 2) + 7 < 5

Solutions

  1. For 4(2x - 1) > 8:

    • Distribute: 8x - 4 > 8
    • Isolate: 8x > 12 → x > 1.5

  2. For 5(x + 3) ≤ 2(3x + 1):

    • Distribute: 5x + 15 ≤ 6x + 2
    • Isolate: 15 - 2 ≤ 6x - 5x → 13 ≤ x → x ≥ 13

  3. For 3(x - 2) + 7 < 5:

    • Distribute: 3x - 6 + 7 < 5
    • Combine: 3x + 1 < 5 → 3x < 4 → x < 4/3

Conclusion

Understanding how to solve inequalities with brackets is crucial for tackling higher-level math problems. By following the steps outlined in this guide, you can break down complex problems into manageable parts. Remember to practice regularly to improve your skills and gain confidence. 😊 Happy solving!

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