Hey guys! Let's dive into the world of inequalities, specifically those involving absolute values. Absolute value inequalities might seem a bit intimidating at first, but with a systematic approach, they become quite manageable. In this guide, we'll tackle two example problems, breaking down each step to ensure you grasp the concepts thoroughly. We'll also emphasize the importance of expressing solutions in interval notation, a standard way to represent sets of real numbers.
Understanding Absolute Value
Before we jump into solving inequalities, let's quickly recap what absolute value means. The absolute value of a number is its distance from zero on the number line. So, |5| = 5 and |-5| = 5. This means that when we have an expression inside absolute value bars, it could be either positive or negative, but its distance from zero is what matters.
Key Concepts for Solving Absolute Value Inequalities
When dealing with absolute value inequalities, there are two main types to consider:
- |x| < a (or |x| ≤ a): This means that the distance of x from zero is less than a. This translates to a compound inequality: -a < x < a (or -a ≤ x ≤ a). Think of it as x being "sandwiched" between -a and a.
- |x| > a (or |x| ≥ a): This means that the distance of x from zero is greater than a. This translates to two separate inequalities: x < -a or x > a (or x ≤ -a or x ≥ a). Think of it as x being "outside" the range between -a and a.
These concepts are crucial for correctly interpreting and solving absolute value inequalities. Now, let's apply these concepts to our example problems.
Example 1: Solving -3 - 7|2x - 9| < -24
Let's start with our first inequality: -3 - 7|2x - 9| < -24. Our goal is to isolate the absolute value expression first. Think of it like solving a regular equation – we want to get the term with the absolute value by itself.
Step 1: Isolate the Absolute Value
To isolate the absolute value term, we need to get rid of the -3. We can do this by adding 3 to both sides of the inequality:
-3 - 7|2x - 9| + 3 < -24 + 3
This simplifies to:
-7|2x - 9| < -21
Now, we need to get rid of the -7 that's multiplying the absolute value. We can do this by dividing both sides by -7. Important: Remember that when we multiply or divide both sides of an inequality by a negative number, we need to flip the inequality sign. So, we have:
-7|2x - 9| / -7 > -21 / -7
This gives us:
|2x - 9| > 3
Great! We've successfully isolated the absolute value expression.
Step 2: Interpret the Absolute Value Inequality
Now we have |2x - 9| > 3. This means the distance of the expression (2x - 9) from zero is greater than 3. Based on our key concepts, this translates into two separate inequalities:
- 2x - 9 < -3
- 2x - 9 > 3
Step 3: Solve Each Inequality
Let's solve each inequality separately.
Inequality 1: 2x - 9 < -3
Add 9 to both sides:
2x - 9 + 9 < -3 + 9
2x < 6
Divide both sides by 2:
2x / 2 < 6 / 2
x < 3
Inequality 2: 2x - 9 > 3
Add 9 to both sides:
2x - 9 + 9 > 3 + 9
2x > 12
Divide both sides by 2:
2x / 2 > 12 / 2
x > 6
So, we have two solutions: x < 3 or x > 6.
Step 4: Express the Solution in Interval Notation
Finally, we need to express our solution in interval notation. Remember that interval notation uses parentheses for strict inequalities (< or >) and brackets for inclusive inequalities (≤ or ≥). Our solution x < 3 represents all numbers less than 3, which in interval notation is (-∞, 3). The solution x > 6 represents all numbers greater than 6, which in interval notation is (6, ∞). Since we have "or" connecting these solutions, we use the union symbol (∪) to combine them.
Therefore, the solution in interval notation is:
(-∞, 3) ∪ (6, ∞)
This means any number in either of these intervals will satisfy the original inequality.
Example 2: Solving 6 - 9|2x + 7| > -3
Now, let's tackle our second inequality: 6 - 9|2x + 7| > -3. We'll follow the same steps as before: isolate the absolute value, interpret the inequality, solve the resulting inequalities, and express the solution in interval notation.
Step 1: Isolate the Absolute Value
First, we need to get rid of the 6. Subtract 6 from both sides:
6 - 9|2x + 7| - 6 > -3 - 6
-9|2x + 7| > -9
Next, divide both sides by -9. Remember to flip the inequality sign since we're dividing by a negative number:
-9|2x + 7| / -9 < -9 / -9
|2x + 7| < 1
We've successfully isolated the absolute value!
Step 2: Interpret the Absolute Value Inequality
Now we have |2x + 7| < 1. This means the distance of the expression (2x + 7) from zero is less than 1. This translates into a compound inequality:
-1 < 2x + 7 < 1
Step 3: Solve the Compound Inequality
To solve this compound inequality, we need to isolate x in the middle. We can do this by performing the same operations on all three parts of the inequality.
First, subtract 7 from all parts:
-1 - 7 < 2x + 7 - 7 < 1 - 7
-8 < 2x < -6
Next, divide all parts by 2:
-8 / 2 < 2x / 2 < -6 / 2
-4 < x < -3
So, our solution is -4 < x < -3.
Step 4: Express the Solution in Interval Notation
Finally, we need to express our solution in interval notation. The inequality -4 < x < -3 represents all numbers between -4 and -3, not including the endpoints. In interval notation, this is written as:
(-4, -3)
This means any number within this interval will satisfy the original inequality.
Key Takeaways for Solving Absolute Value Inequalities
- Isolate the absolute value expression: This is the first crucial step. Get the |expression| by itself on one side of the inequality.
- Interpret the inequality: Remember the two main cases:
- |x| < a translates to -a < x < a
- |x| > a translates to x < -a or x > a
- Solve the resulting inequalities: Solve each inequality separately, whether it's a compound inequality or two separate inequalities.
- Express the solution in interval notation: This is the standard way to represent the solution set for inequalities.
- Remember to flip the inequality sign: When multiplying or dividing by a negative number, flip the inequality sign.
Practice Makes Perfect
Solving absolute value inequalities requires practice. The more you work through examples, the more comfortable you'll become with the process. Try solving various absolute value inequalities, and don't hesitate to review the steps outlined in this guide whenever you need a refresher.
By understanding the core concepts and following a systematic approach, you can confidently tackle any absolute value inequality that comes your way. Keep practicing, and you'll master this important skill in no time! Remember, patience and persistence are key!