What is the solution for the following absolute value inequality?
Absolute value inequalities can be solved by considering two cases: when the value inside the absolute value symbol is positive and when it is negative. Let’s solve an example to understand the process.
Consider the inequality: |3x – 4| ≤ 7
To solve this, we need to divide it into two cases:
Case 1: (3x – 4) ≥ 0
When the expression inside the absolute value is non-negative, we can remove the absolute value symbols without changing the inequality sign. Therefore, in this case, we have:
3x – 4 ≤ 7
Solving this linear inequality:
3x ≤ 11
x ≤ 11/3
Case 2: (3x – 4) < 0
When the expression inside the absolute value is negative, we need to multiply the inequality sign by -1 and flip the inequality direction. Thus, in this case, we have:
-(3x – 4) ≤ 7
Simplifying:
-3x + 4 ≤ 7
-3x ≤ 3
x ≥ -1
Combining the solutions from both cases:
The solution for the absolute value inequality |3x – 4| ≤ 7 is -1 ≤ x ≤ 11/3.
FAQs:
1. What is an absolute value inequality?
An absolute value inequality involves an absolute value expression (such as |x|) and a relational inequality symbol (such as ≤ or ≥).
2. How do you solve an absolute value inequality with a positive expression inside?
When the expression inside the absolute value is non-negative, you can remove the absolute value symbols without changing the inequality sign.
3. How do you solve an absolute value inequality with a negative expression inside?
When the expression inside the absolute value is negative, multiply the inequality sign by -1 and flip the inequality direction.
4. Can absolute value inequalities have more than one solution?
Yes, absolute value inequalities can have multiple solutions. The solution to an absolute value inequality often consists of a range of values.
5. What happens if the inequality symbol is strict, such as < or >?
If the inequality symbol is strict, such as < or >, the solution will not include the boundary points. They will be excluded from the solution.
6. Are the solutions to absolute value inequalities always real numbers?
Yes, the solutions to absolute value inequalities are always real numbers, unless the inequality is impossible to satisfy.
7. Can absolute value inequalities be solved using graphical methods?
Yes, absolute value inequalities can indeed be solved graphically. The graph of the absolute value expression can help determine the solution range.
8. Are there any cases where absolute value inequalities have no solution?
Yes, there are cases where absolute value inequalities have no solution, such as when the absolute value expression is less than zero and the inequality symbol is ≥ or ≤.
9. Can the solution to an absolute value inequality be an empty set?
Yes, it is possible for the solution to an absolute value inequality to be an empty set if the inequality cannot be satisfied under any circumstances.
10. Can the solution to an absolute value inequality be an infinite set?
No, the solution to an absolute value inequality is always a finite set or an empty set. It cannot be an infinite set.
11. What if the absolute value inequality contains multiple absolute value expressions?
If an absolute value inequality contains multiple absolute value expressions, each expression should be treated separately, considering the conditions for both positive and negative values.
12. Are there any shortcuts or tricks to solve absolute value inequalities more easily?
No specific shortcuts or tricks apply to all absolute value inequalities, as each one may have different conditions and expressions. It is important to carefully analyze and solve each inequality on its own merits.