Handling absolute value inequalities can often feel like a puzzle, but understanding the rules and applying systematic approaches can make solving them straightforward and rewarding. Let's delve into a detailed guide on how to navigate these inequalities, complete with examples and answers for a solid grasp of the concept.
Understanding Absolute Value Inequalities
Before we jump into the examples, let’s ensure you’re familiar with absolute value inequalities:
- |x| < a: This means x must be between -a and a.
- |x| > a: Here, x is either less than -a or greater than a.
- |x| ≤ a: Similar to the less than, x can be equal to a or -a or any number between them.
- |x| ≥ a: x can be equal to a, less than -a, or greater than a.
Example 1: Solving |x - 3| < 5
This inequality represents:
- x - 3 < 5 and
- x - 3 > -5
Now let’s solve these two inequalities:
- x < 8
- x > -2
So, the solution is:
-2 < x < 8
Example 2: Solving |x + 4| ≥ 2
This translates into two inequalities:
- x + 4 ≥ 2 or
- x + 4 ≤ -2
Solving these, we get:
- x ≥ -2
- x ≤ -6
Therefore, the solution set is:
x ≤ -6 or x ≥ -2
🧐 Note: When dealing with "or" statements, the solution includes all values satisfying either condition.
Example 3: Solving |2x + 3| < 11
Again, this means:
- 2x + 3 < 11 and
- 2x + 3 > -11
Solving these, we find:
- x < 4
- x > -7
So, the solution is:
-7 < x < 4
Example 4: Solving |x - 1| ≥ 6
This inequality becomes:
- x - 1 ≥ 6 or
- x - 1 ≤ -6
After solving:
- x ≥ 7
- x ≤ -5
Thus, the solution set is:
x ≤ -5 or x ≥ 7
👉 Note: Notice how solutions diverge when dealing with 'or' statements, showing intervals separate from each other.
Graphing Solutions
When graphing solutions, the following rules apply:
- Use open circles for less than or greater than (<, >).
- Use closed circles for less than or equal to or greater than or equal to (≤, ≥).
- For inequalities with “or,” draw two separate lines.
- For inequalities with “and,” show a single continuous line.
| Inequality | Solution Set | Graph Description |
|---|---|---|
| |x - 3| < 5 | -2 < x < 8 | Single continuous line between -2 and 8 (open circles at endpoints). |
| |x + 4| ≥ 2 | x ≤ -6 or x ≥ -2 | Two separate lines, one from negative infinity to -6 (closed circle), another from -2 to infinity (open circle). |
| |2x + 3| < 11 | -7 < x < 4 | Single continuous line between -7 and 4 (open circles at endpoints). |
| |x - 1| ≥ 6 | x ≤ -5 or x ≥ 7 | Two separate lines, one from negative infinity to -5 (closed circle), another from 7 to infinity (open circle). |
In wrapping up, solving absolute value inequalities involves understanding how to interpret the absolute value notation in terms of ranges or sets of numbers. By breaking down the inequalities into more manageable parts and then applying the basic inequality rules, you'll master this algebra skill. Remember to pay attention to whether the inequality includes equality signs, as this affects how you represent solutions graphically.
What does it mean when an absolute value inequality includes an “or”?
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An “or” in absolute value inequalities means that the solution can satisfy either of the conditions derived from the inequality. Graphically, this is represented by two separate intervals, indicating all values where the conditions are true.
How do you handle absolute values with a greater than or equal to sign?
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For inequalities with “≥” or “≤”, you solve for both greater than and less than conditions, and the solutions include the endpoints themselves, shown with closed circles on the graph.
Can you explain how the equal sign in absolute value inequalities affects the solutions?
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The equal sign means that the solution includes the point(s) where the absolute value equals the given number. This impacts the graph by using closed circles to denote inclusion of those points in the solution set.
