Solving Absolute Value Inequalities: A Step-by-Step Guide

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Hey math whizzes and anyone else who's ever stared at an absolute value inequality and thought, "What the heck am I supposed to do with this?" Today, we're diving deep into solving inequalities like the one you threw at us: $|2 x-7|+18 ≥ 30$. Don't worry, guys, it's not as scary as it looks! We're going to break it down, step by step, so you can tackle these bad boys with confidence. So, grab your notebooks, get comfy, and let's get this math party started!

Understanding Absolute Value

Before we jump into solving, let's quickly chat about what absolute value actually is. Think of the absolute value of a number as its distance from zero on the number line. It's always a positive number (or zero). For example, the absolute value of 5 is 5, and the absolute value of -5 is also 5. We write it using two vertical bars, like this: ∣x∣|x|. So, ∣5∣=5|5| = 5 and ∣−5∣=5|-5| = 5. When we see an expression inside absolute value bars, like ∣2x−7∣|2x-7|, it means we're interested in the distance of that expression from zero. This is key because it tells us there are usually two possibilities to consider when solving absolute value equations or inequalities.

Isolating the Absolute Value Expression

Our first mission, should we choose to accept it (and we totally should!), is to get that absolute value expression all by itself on one side of the inequality. In our example, $|2 x-7|+18 ≥ 30$, we need to get rid of that '+18'. How do we do that? By doing the opposite, of course! We'll subtract 18 from both sides of the inequality. Remember, whatever you do to one side, you must do to the other to keep the inequality balanced. So, we have:

∣2x−7∣+18−18≥30−18|2 x-7|+18 - 18 ≥ 30 - 18

This simplifies to:

∣2x−7∣≥12|2 x-7| ≥ 12

See? Now we have the absolute value expression isolated. This is a crucial step because it sets us up for the next part of the puzzle.

The Two-Case Scenario

Here's where the magic of absolute value really comes into play. Because the expression inside the absolute value bars, $2x-7$, can be either positive or negative, we need to consider both possibilities that result in it being greater than or equal to 12.

Case 1: The expression inside is positive (or zero).

If $2x-7$ is positive or zero, then its absolute value is just the expression itself. So, we can simply write:

2x−7≥122x-7 ≥ 12

Case 2: The expression inside is negative.

If $2x-7$ is negative, then its absolute value is the opposite of the expression. To make the inequality true, the expression itself must be less than or equal to the negative of the value on the other side. So, we write:

2x−7≤−122x-7 ≤ -12

Notice how the inequality sign flips in the second case when we remove the absolute value. This is super important! We now have two separate, simpler linear inequalities to solve. It's like we've turned one tricky problem into two manageable ones. Pretty neat, right?

Solving Case 1

Let's tackle the first inequality: $2x-7 ≥ 12$.

Again, we want to get 'x' all by itself. First, we add 7 to both sides:

2x−7+7≥12+72x - 7 + 7 ≥ 12 + 7

2x≥192x ≥ 19

Now, to isolate 'x', we divide both sides by 2:

2x/2≥19/22x / 2 ≥ 19 / 2

x≥9.5x ≥ 9.5

So, one part of our solution is that 'x' must be greater than or equal to 9.5. Keep this in mind!

Solving Case 2

Now, let's solve the second inequality: $2x-7 ≤ -12$.

Similar to the first case, we start by adding 7 to both sides:

2x−7+7≤−12+72x - 7 + 7 ≤ -12 + 7

2x≤−52x ≤ -5

And now, we divide both sides by 2 to get 'x' alone:

2x/2≤−5/22x / 2 ≤ -5 / 2

x≤−2.5x ≤ -2.5

And there we have it! The second part of our solution is that 'x' must be less than or equal to -2.5.

Combining the Solutions

We've solved both cases, and now we need to bring our findings together. Remember, the original inequality $|2 x-7|+18 ≥ 30$ is true if either Case 1 or Case 2 is true. This means our solution is the combination of both results.

We found that $x ≥ 9.5$ AND $x ≤ -2.5$.

On a number line, this looks like two separate intervals. For $x ≤ -2.5$, we're talking about all the numbers to the left of -2.5, including -2.5 itself. For $x ≥ 9.5$, we're talking about all the numbers to the right of 9.5, including 9.5 itself.

We can write this solution in interval notation. The interval for $x ≤ -2.5$ is (−∞,−2.5](-\infty, -2.5]. The interval for $x ≥ 9.5$ is [9.5,∞)[9.5, \infty).

Since the solution includes either of these possibilities, we combine them using the union symbol (∪).

So, the final solution to the inequality $|2 x-7|+18 ≥ 30$ is:

x≤−2.5 or x≥9.5x ≤ -2.5 \text{ or } x ≥ 9.5

Or in interval notation:

(−∞,−2.5]∪[9.5,∞)(-\infty, -2.5] \cup [9.5, \infty)

Verification: Does it Work?

Math folks, we're not done yet! A good habit to get into is verifying your solution. Let's pick a value from each part of our solution and plug it back into the original inequality $|2 x-7|+18 ≥ 30$ to make sure it holds true.

Test a value from $x ≤ -2.5$: Let's try x=−3x = -3.

∣2(−3)−7∣+18≥30|2(-3) - 7| + 18 ≥ 30

∣−6−7∣+18≥30|-6 - 7| + 18 ≥ 30

∣−13∣+18≥30|-13| + 18 ≥ 30

13+18≥3013 + 18 ≥ 30

31≥3031 ≥ 30

This is TRUE! Awesome.

Test a value from $x ≥ 9.5$: Let's try x=10x = 10.

∣2(10)−7∣+18≥30|2(10) - 7| + 18 ≥ 30

∣20−7∣+18≥30|20 - 7| + 18 ≥ 30

∣13∣+18≥30|13| + 18 ≥ 30

13+18≥3013 + 18 ≥ 30

31≥3031 ≥ 30

This is also TRUE! Fantastic.

Test a value not in our solution: Let's try x=0x = 0 (which is between -2.5 and 9.5).

∣2(0)−7∣+18≥30|2(0) - 7| + 18 ≥ 30

∣0−7∣+18≥30|0 - 7| + 18 ≥ 30

∣−7∣+18≥30|-7| + 18 ≥ 30

7+18≥307 + 18 ≥ 30

25≥3025 ≥ 30

This is FALSE. Perfect! It confirms that values between -2.5 and 9.5 are indeed not part of our solution.

Key Takeaways for Absolute Value Inequalities

So, what did we learn, guys? Solving absolute value inequalities boils down to a few key steps:

  1. Isolate the absolute value expression: Get the ∣...∣|...| part all by itself on one side.
  2. Split into two cases: Because absolute value means distance, the expression inside can be positive or negative. This creates two separate inequalities.
    • For ∣expression∣≥c|expression| ≥ c, you get expression≥cexpression ≥ c OR expression≤−cexpression ≤ -c.
    • For ∣expression∣≤c|expression| ≤ c, you get expression≤cexpression ≤ c AND expression≥−cexpression ≥ -c.
  3. Solve each inequality: Treat them like regular linear inequalities.
  4. Combine the solutions: Use 'OR' for '≥' and '≤' inequalities, and 'AND' for '<' and '>' inequalities. Represent your final answer using inequality notation or interval notation.
  5. Verify your solution: Plug in test values to make sure everything checks out.

Mastering these steps will make solving any absolute value inequality a breeze. Remember, practice makes perfect! Keep working through these problems, and you'll be an absolute value pro in no time. Happy solving!