Solve The Inequality 4x - 4 ≤ 2: A Step-By-Step Guide To Mastering Inequalities

Solving inequalities is a crucial skill that opens doors to understanding a wide variety of mathematical concepts. Inequalities help us express ranges of values and can be incredibly useful in real-life applications like budgeting, engineering, and statistics. One of the simplest inequalities you might encounter is of the form (4x - 4 \leq 2). In this guide, we’ll break it down step-by-step so you can master similar inequalities with confidence.

Understanding the Inequality

The inequality we’re working with is:

[4x - 4 \leq 2]

This means that the expression (4x - 4) is less than or equal to 2. Our goal is to isolate (x) and find all values that satisfy this condition.

Step 1: Add 4 to Both Sides

To begin, we want to simplify the expression by eliminating the constant term on the left side. We do this by adding 4 to both sides of the inequality:

[4x - 4 + 4 \leq 2 + 4]

This simplifies to:

[4x \leq 6]

Step 2: Divide by 4

Now that we have (4x) isolated, we will divide both sides by 4 to solve for (x):

[\frac{4x}{4} \leq \frac{6}{4}]

This reduces to:

[x \leq \frac{3}{2}]

Step 3: Interpret the Solution

The solution (x \leq \frac{3}{2}) means that (x) can take any value that is less than or equal to (1.5). You can also express this in interval notation as:

[ (-\infty, 1.5] ]

Visual Representation

To better understand the solution, it’s useful to visualize it on a number line. Here's a simple representation:

<---|-------------------|-----|------>
    -1                  0    1.5

The shaded region to the left of (1.5) (including (1.5) itself) indicates all possible values for (x).

Common Mistakes to Avoid

1. Ignoring the Inequality Sign: Always remember that when you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign.

2. Not Checking the Solution: After finding your solution, it's good practice to plug a value back into the original inequality to ensure it holds true. For instance, testing (x = 1):

   [4(1) - 4 \leq 2 \implies 0 \leq 2] (True)

3. Overcomplicating the Steps: Simplifying the process by carefully adding or subtracting terms can save time and reduce errors.

Troubleshooting Tips

If you find yourself stuck or unsure about an inequality, consider the following:

• Revisit Basic Algebra: Make sure you’re comfortable with basic algebraic operations.

• Work Through Examples: Practice with different inequalities to recognize patterns.

• Break it Down: If the inequality has multiple terms, tackle one term at a time.

Example Problem

Let’s work through another example:

Solve the inequality (2x + 5 > 9)

1. Subtract 5 from both sides: [ 2x > 4 ]

2. Divide by 2: [ x > 2 ]

The solution is (x > 2), which in interval notation is ((2, \infty)).

Frequently Asked Questions

<div class="faq-section"> <div class="faq-container"> <h2>Frequently Asked Questions</h2> <div class="faq-item"> <div class="faq-question"> <h3>What is the difference between equations and inequalities?</h3> <span class="faq-toggle">+</span> </div> <div class="faq-answer"> <p>Equations state that two expressions are equal, while inequalities express a range of values that satisfy a particular condition (e.g., less than, greater than).</p> </div> </div> <div class="faq-item"> <div class="faq-question"> <h3>Can an inequality have multiple solutions?</h3> <span class="faq-toggle">+</span> </div> <div class="faq-answer"> <p>Yes, most inequalities have a range of solutions rather than a single value, as seen with the example of (x \leq \frac{3}{2}).</p> </div> </div> <div class="faq-item"> <div class="faq-question"> <h3>How do I graph inequalities?</h3> <span class="faq-toggle">+</span> </div> <div class="faq-answer"> <p>To graph an inequality, draw a number line, shade the appropriate region (all values that satisfy the inequality), and use an open circle for strict inequalities and a closed circle for inclusive inequalities.</p> </div> </div> </div> </div>

Recap of the key takeaways: we have thoroughly explored how to solve the inequality (4x - 4 \leq 2). Through adding, dividing, and interpreting our results, you’ve learned that the values of (x) satisfying this inequality lie below or equal to (1.5). We’ve also shared tips to avoid common pitfalls and offered advice on troubleshooting. The world of inequalities is fascinating, so keep practicing!

<p class="pro-note">🔍Pro Tip: Always check your solution by plugging values back into the original inequality!</p>