How do you know if an absolute value inequality is "No Solution" or not?
You have to try to solve the inequality and then conclude at some point that there are no solutions. Sometimes you can use some inequalities that always hold, like:
|a| >= 0
|a+b| <= |a| + |b|
etc.
If you can rearrange an inequality so that it contradicts these standard inequalities, it cannot have a solution.
Well, when it comes to absolute value inequalities, it's like playing a game of hide-and-seek. If the absolute value of a number is greater than a certain value, then you know that number is hiding out beyond that range. But if the absolute value of a number is less than or equal to the given value, then you know that number is willing to come out and play within that range. So, if you end up with an absolute value inequality that has no intersection between the solution set and the number line, it's like looking for Waldo and realizing he's taken a vacation. That's when you can confidently say there's "No Solution" to the absolute value inequality.
To determine if an absolute value inequality has no solution, you need to examine the inequality and analyze its properties. Follow these steps:
Step 1: Rewrite the inequality with an absolute value expression on the left side of the inequality symbol.
Step 2: Identify the inequality symbol (>, <, ≥, or ≤).
Step 3: Set up two separate cases, one without the absolute value and one with the absolute value.
Step 4: Without the absolute value, rewrite the inequality as two separate inequalities, one using a positive constant and the other using a negative constant. Eliminate the absolute value symbol.
Step 5: Solve each inequality separately from step 4.
Step 6: Analyze and compare the solution sets from steps 4 and 5.
If the solution set from steps 4 and 5 both have no solutions, then the absolute value inequality overall has no solution.
On the other hand, if there is a solution in either case, then the absolute value inequality has at least one solution.
To determine if an absolute value inequality has no solution, you need to consider the given inequality and apply the properties of absolute values. Here's how you can determine if there is no solution:
1. Start by isolating the absolute value expression on one side of the inequality. For example, if you have an inequality like |x + 3| < 2, subtract 3 from both sides to get |x| < 2 - 3, which simplifies to |x| < -1.
2. Remember that the absolute value of any number is always non-negative, meaning it is either zero or positive. Consequently, the expression |x| < -1 has no solution because there are no non-negative numbers that are less than -1.
3. Therefore, if the absolute value inequality results in an expression that is always less than zero, such as |x| < -1, it has no solution. In other words, when you find that the inequality leads to an impossible situation where the absolute value is less than a negative number, no values of x will satisfy the inequality.
It is essential to observe and conclude carefully to determine if an absolute value inequality has no solution.