Is the solution to 0 ≤ |4u - 7| u ≥ 7/4?

The absolute value of any real function is always greater than or equal to zero, by definition of the absolute value function.

Therefore the solution (domain) of the inequality is ℝ.

We can work it out as follows:
Rewrite
0 ≤ |4u-7|
as
|4u-7| ≥ 0

Transform the absolute function into two inequalities:

4u-7 ≥0 when 4u-7≥0 ...(1)
-(4u-7) ≥0 when 4u-7<0...(2)

For (1)
4u-7≥0
4u ≥ 7
u ≥ 7/4 if u≥ 7/4

For (2)
If u < 7/4,
-(4u-7) > 0
4u-7<<0
4u < 7
u < 7/4

Which means that the answer to the inequality is:
u=(-∞,∞)

thanks

You're welcome!

To determine if the solution to the inequality 0 ≤ |4u - 7| is u ≥ 7/4, we need to solve the inequality and then check if the solution satisfies the given condition.

First, let's solve the inequality |4u - 7| ≥ 0. The absolute value of any number is always non-negative, so the absolute value of (4u - 7) can only be equal to or greater than zero. Therefore, we can conclude that |4u - 7| ≥ 0 is always true.

Now, let's check if the solution u ≥ 7/4 satisfies the given condition. When we substitute u = 7/4 into the equation |4u - 7|, we get:
|4(7/4) - 7| = |7 - 7| = |0| = 0

Since the absolute value of 0 is 0 and 0 is greater than or equal to 0, the given condition u ≥ 7/4 is satisfied. Therefore, the solution to the inequality 0 ≤ |4u - 7| is u ≥ 7/4.