Mastering Absolute Value: Equations & Inequalities

Hey math enthusiasts, welcome back to Plastik Magazine! Today, we're diving deep into the sometimes tricky, but totally conquerable, world of absolute value. Whether you're wrestling with equations or trying to tame inequalities, understanding absolute value is a superpower you'll want in your mathematical toolkit. So, grab your calculators, dust off those notebooks, and let's break down how to solve some common absolute value problems like a boss!

Part A: Solving Absolute Value Equations - The Breakdown

Alright guys, let's tackle the first beast: solving the equation |5x + 4| = 3|x + 9|. When you see absolute values on both sides of an equation, it means we have a couple of scenarios to consider because the expression inside the absolute value bars could be positive or negative.

The core idea behind absolute value is distance from zero. So, if |A| = B, it means A is either equal to B or A is equal to -B. When we have two absolute value expressions, like |A| = |B|, we need to consider two main cases: either A = B or A = -B. However, in our specific problem, |5x + 4| = 3|x + 9|, we have a coefficient in front of one of the absolute values. This means we need to handle the '3' as well. The fundamental principle remains: the expression on the left, 5x + 4, must be equal to either positive 3|x + 9| or negative 3|x + 9|. This expands into two distinct equations we need to solve.

Case 1: The Positive Path

First, let's assume that the expressions inside the absolute value bars are related such that 5x + 4 = 3(x + 9). This is our first scenario. To solve this, we simply distribute the 3 on the right side:

5x + 4 = 3x + 27

Now, we gather our 'x' terms on one side and our constants on the other. Subtract 3x from both sides:

5x - 3x + 4 = 27

2x + 4 = 27

Next, subtract 4 from both sides:

2x = 27 - 4

2x = 23

Finally, divide by 2:

x = 23/2

So, one potential solution is x = 23/2. Keep this in your back pocket!

Case 2: The Negative Path

Now, for our second scenario, we consider the case where 5x + 4 = -3(x + 9). Again, the first step is to distribute the -3 on the right side:

5x + 4 = -3x - 27

Let's bring the 'x' terms together. Add 3x to both sides:

5x + 3x + 4 = -27

8x + 4 = -27

Now, isolate the 'x' term by subtracting 4 from both sides:

8x = -27 - 4

8x = -31

And finally, divide by 8:

x = -31/8

Our second potential solution is x = -31/8.

Verification (Always a Good Idea!)

It's crucial with absolute value equations to verify your solutions by plugging them back into the original equation. This ensures that we haven't introduced any extraneous solutions.

• Checking x = 23/2:

  • Left side: |5(23/2) + 4| = |115/2 + 8/2| = |123/2| = 123/2
  • Right side: 3|23/2 + 9| = 3|23/2 + 18/2| = 3|41/2| = 3 * (41/2) = 123/2
  • The left side equals the right side. x = 23/2 is a valid solution.

• Checking x = -31/8:

  • Left side: |5(-31/8) + 4| = |-155/8 + 32/8| = |-123/8| = 123/8
  • Right side: 3|-31/8 + 9| = 3|-31/8 + 72/8| = 3|41/8| = 3 * (41/8) = 123/8
  • The left side equals the right side. x = -31/8 is a valid solution.

So, the solutions for |5x + 4| = 3|x + 9| are x = 23/2 and x = -31/8. Nicely done!

Part B: Tackling Absolute Value Inequalities

Now, let's level up and solve the inequality |(3x - 4)/5| - 3 ≥ 2. Inequalities involving absolute values require a slightly different approach, but the core principles of absolute value still apply. Our goal here is to isolate the absolute value expression first, just like we'd isolate a variable in a regular inequality.

Step 1: Isolate the Absolute Value

First things first, let's get that absolute value term by itself on one side of the inequality. We can do this by adding 3 to both sides:

|(3x - 4)/5| - 3 + 3 ≥ 2 + 3

|(3x - 4)/5| ≥ 5

We've now isolated the absolute value. This inequality tells us that the expression (3x - 4)/5 must be either greater than or equal to 5, OR less than or equal to -5. Think about it: the distance of (3x - 4)/5 from zero must be at least 5 units.

Step 2: Split into Two Inequalities

Just like with equations, we split this into two separate inequalities, keeping the 'greater than or equal to' and 'less than or equal to' signs in mind:

Inequality 1: The expression inside is greater than or equal to 5.

(3x - 4)/5 ≥ 5

Inequality 2: The expression inside is less than or equal to -5.

(3x - 4)/5 ≤ -5

Step 3: Solve Each Inequality

Let's solve Inequality 1:

(3x - 4)/5 ≥ 5

Multiply both sides by 5 to clear the denominator:

3x - 4 ≥ 25

Add 4 to both sides:

3x ≥ 29

Divide by 3:

x ≥ 29/3

Now, let's solve Inequality 2:

(3x - 4)/5 ≤ -5

Multiply both sides by 5:

3x - 4 ≤ -25

Add 4 to both sides:

3x ≤ -21

Divide by 3:

x ≤ -7

Step 4: Combine and Express in Interval Notation

We have two conditions that satisfy the original inequality: x ≥ 29/3 OR x ≤ -7. These are two separate ranges of values for x.

• The condition x ≤ -7 means all numbers less than or equal to -7. In interval notation, this is (-∞, -7]. Remember the bracket indicates that -7 is included.
• The condition x ≥ 29/3 means all numbers greater than or equal to 29/3. In interval notation, this is [29/3, ∞). The bracket indicates that 29/3 is included.

Since the solution is EITHER x ≤ -7 OR x ≥ 29/3, we combine these two intervals using the union symbol (∪).

Therefore, the solution to the inequality |(3x - 4)/5| - 3 ≥ 2 in interval notation is (-∞, -7] ∪ [29/3, ∞).

And there you have it, legends! We've successfully navigated both absolute value equations and inequalities. Remember, the key is to understand what absolute value represents (distance) and to carefully consider all possible cases, especially when dealing with equations. For inequalities, isolating the absolute value and then splitting into two separate conditions is your golden ticket. Keep practicing, and you'll be an absolute value whiz in no time! Stay curious, and we'll catch you in the next math adventure here at Plastik Magazine!