Solving Absolute Value Inequality: |3x + 1| ≤ 5

Hey Plastik Magazine readers! Today, we're diving into the fascinating world of absolute value inequalities. Specifically, we're going to break down how to solve the inequality |3x + 1| ≤ 5. This type of problem might seem a bit intimidating at first, but trust me, it's totally manageable once you understand the core concepts. We'll walk through it step by step, making sure you grasp every detail. So, grab your thinking caps, and let's get started!

Understanding Absolute Value

Before we jump into solving the inequality, let's quickly recap what absolute value means. Absolute value, denoted by |x|, represents the distance of a number x from zero on the number line. Because distance is always non-negative, the absolute value of a number is always positive or zero. For example, |3| = 3 and |-3| = 3. This concept is crucial for understanding how to deal with inequalities involving absolute values.

The key takeaway here is that an absolute value expression essentially creates two possibilities: the expression inside the absolute value can be either positive or negative, but its distance from zero remains the same. This duality is what makes solving absolute value inequalities a bit different from solving regular inequalities.

The Golden Rule of Absolute Value Inequalities

When dealing with inequalities like |3x + 1| ≤ 5, we need to remember a crucial rule: If |a| ≤ b, then -b ≤ a ≤ b. This rule is the cornerstone of solving these types of problems. It allows us to transform a single absolute value inequality into a compound inequality, which is much easier to handle. Keep this rule in your mental toolkit; it's going to be your best friend for this type of problem!

Breaking Down the Inequality |3x + 1| ≤ 5

Now that we've refreshed our understanding of absolute value, let's tackle the problem at hand: |3x + 1| ≤ 5. Applying the rule we just discussed, we can rewrite this absolute value inequality as a compound inequality. This means we need to consider both the positive and negative possibilities of the expression inside the absolute value.

Transforming into a Compound Inequality

Using the rule |a| ≤ b ⟺ -b ≤ a ≤ b, we can rewrite |3x + 1| ≤ 5 as: -5 ≤ 3x + 1 ≤ 5. See how we've transformed the single absolute value inequality into a more manageable form? This compound inequality essentially states that the expression 3x + 1 must fall between -5 and 5, inclusive. It's like saying 3x + 1 is trapped within this range.

Solving the Compound Inequality

Our next step is to solve the compound inequality -5 ≤ 3x + 1 ≤ 5. To do this, we need to isolate x in the middle. This involves performing the same operations on all three parts of the inequality: the left side, the middle, and the right side. Think of it as maintaining balance on a scale; whatever you do to one side, you must do to the others.

Isolating x

First, let's subtract 1 from all parts of the inequality: -5 - 1 ≤ 3x + 1 - 1 ≤ 5 - 1, which simplifies to -6 ≤ 3x ≤ 4. Now, we're one step closer to isolating x. Next, we'll divide all parts of the inequality by 3: -6/3 ≤ 3x/3 ≤ 4/3, which simplifies to -2 ≤ x ≤ 4/3. Voila! We've successfully isolated x, giving us the range of values that satisfy the original inequality.

Identifying Solutions from the Given Options

We've determined that the solutions to the inequality |3x + 1| ≤ 5 must fall within the range -2 ≤ x ≤ 4/3. Now, let's look at the options provided and see which ones fit within this range. This is where our hard work pays off, and we get to pick the correct answers.

Evaluating the Options

The options given are: A) -2, B) 1, C) 2, D) 0, E) -3. Let's evaluate each one:

• A) -2: Since -2 ≤ -2 ≤ 4/3, this is a solution.
• B) 1: Since -2 ≤ 1 ≤ 4/3 (which is approximately 1.33), this is also a solution.
• C) 2: Since 2 is greater than 4/3, this is not a solution.
• D) 0: Since -2 ≤ 0 ≤ 4/3, this is a solution.
• E) -3: Since -3 is less than -2, this is not a solution.

The Solutions: A, B, and D

After carefully evaluating each option, we've found that the solutions to the inequality |3x + 1| ≤ 5 from the given choices are A) -2, B) 1, and D) 0. Congratulations, guys! You've successfully navigated the world of absolute value inequalities. Remember, the key is to understand the fundamental rule and break down the problem into manageable steps.

Common Pitfalls and How to Avoid Them

Solving absolute value inequalities can sometimes be tricky, and it's easy to make common mistakes. Let's look at some of these pitfalls and how to avoid them. Being aware of these potential errors can help you ace these types of problems every time.

Forgetting the Negative Case

One of the most common mistakes is forgetting to consider the negative case when dealing with absolute values. Remember, the expression inside the absolute value can be either positive or negative, and both cases need to be accounted for. Always rewrite the absolute value inequality as a compound inequality to ensure you cover both possibilities. For example, for |3x + 1| ≤ 5, don't just think about 3x + 1 ≤ 5; remember to also consider -5 ≤ 3x + 1.

Incorrectly Applying the Rule

Another pitfall is misapplying the rule for absolute value inequalities. Make sure you understand the rule |a| ≤ b ⟺ -b ≤ a ≤ b thoroughly. It's easy to get confused with the signs or the direction of the inequality. Practice applying this rule with different examples to solidify your understanding.

Arithmetic Errors

Simple arithmetic errors can also lead to incorrect solutions. When solving the compound inequality, be careful with your addition, subtraction, multiplication, and division. Double-check each step to ensure accuracy. It's a good idea to write out each step clearly to minimize the chances of making a mistake.

Not Checking Solutions

Finally, a crucial step that's often overlooked is checking your solutions. Once you've found potential solutions, plug them back into the original inequality to make sure they satisfy it. This is a quick way to catch any errors and ensure your answers are correct. In our case, we checked each option against the range -2 ≤ x ≤ 4/3.

Practice Makes Perfect

The best way to master solving absolute value inequalities is through practice. The more problems you solve, the more comfortable you'll become with the process. Try different variations of these problems, and don't be afraid to challenge yourself. Remember, each problem is an opportunity to learn and improve. Keep practicing, and you'll become a pro in no time!

Real-World Applications

You might be wondering,