Is this correct?

Question

Question: why does the inequality -1/2 x + 1 > | x – 5 | have no solution?

My answer:
the inequality -1/2 x + 1 > | x – 5 | because no number can make the inequality true. This is because only a negative value on the right side of the equation could make it true but since it is an absolute it can not be a negative number.

Answer 1

(-1/2)x + 1 > |x - 5|

This is equivalent to

|x - 5| < (-1/2)x + 1

Absolute inequalities in the form |y| < a implies two inequalities:
y < a AND y > -a. Therefore,

(x - 5) < (-1/2)x + 1 AND
(x - 5) > - [(-1/2)x + 1]

Let's solve each inequality individually.

1) x - 5 < (-1/2)x + 1
x + (1/2)x < 5 + 1
(3/2)x < 6
x < (2/3)(6)
x < (12/3)
x < 4

2) (x - 5) > - [(-1/2)x + 1]

(x - 5) > (1/2)x - 1
x - 5 > (1/2)x - 1
x - (1/2)x > 5 - 1
(1/2)x > 4
x > 6

This means x < -4 AND x > 6. There is no real number that satisfies these two conditions. That's why the inequality
(-1/2)x + 1 > |x - 5| has no solution.

Answer 2

But |x-5| is a bigger term than 1-1/2x for all values of x.
It is wrong, should point the other way.

let x=1 then -1/2+1 =1/2 |1-5| = 4

4>1/2

It could have solution. I think



Regards.

Answer 3

If you graph y=-x/2 +1 and y= |x-5| on the same coordinate system, you will clearly see that |x-5| is everywhere greater than -x/2 +1.