Find an equivalent inequality with absolute value
#1: -5 < y < 5
#2: x < -4 or 4 < x
I'm not sure at all how to solve this problem. Please walk me through the process.
Thanks!
To find an equivalent inequality with absolute value, you need to determine the range of values that satisfy the original inequality condition. Let's go through each inequality one by one:
#1: -5 < y < 5
To express this inequality using absolute value, you want to consider the distance between y and 0. Remember that absolute value measures the distance from a number to zero on a number line.
In this case, we observe that the range of values that satisfy the condition is between -5 and 5, excluding -5 and 5 themselves. Thus, the absolute value inequality equivalent to this is:
|y| < 5
This means that the absolute value of y is less than 5.
#2: x < -4 or 4 < x
This inequality has two separate inequalities connected by the logical operator "or." To express each inequality with absolute value, you need to consider the distance between x and the two given values: -4 and 4.
For the first inequality, x < -4, we can observe that the values satisfying this condition are less than -4. To express this using absolute value, we negate (-) the lower bound value, which is -4, and place it inside the absolute value symbols. This gives us:
|x - (-4)| > 0
For the second inequality, 4 < x, the values satisfying this condition are greater than 4. To express this using absolute value, we again negate (-) the lower bound value and place it inside the absolute value symbols:
|x - 4| > 0
Now, we combine the two inequalities using "or" logic, which means that either of the inequalities should hold true. So the equivalent absolute value inequality is:
|x - (-4)| > 0 or |x - 4| > 0
This means that the absolute value of x minus -4 is greater than 0, or the absolute value of x minus 4 is also greater than 0.
I hope this explanation helps you understand how to find an equivalent inequality with absolute value!