Question

How do you solve the inequality `abs(2x-4)>=6`?

Answer 1

x `>=`5

Explanation:

Move like terms and solve.

1. 2x - 4 `>=`6+4
2. x `>=` `10/2`
3. x `>=`5

Answer 2

`x in ]-oo;-1] uu [5;oo[`

Explanation:

in this case

`abs(a)>=b` then `a<=-b and a>=b`

then in `abs(2x-4)>=6`

`2x-4<=-6` and `2x-4>=6`

`2x-4+6<=0` and `2x-4-6>=0`

`2x+2<=0` and `2x-10>=0`

`x<=-1` and `x>=5`

then `x in]-oo;-1] uu [5;oo[`

Answer 3

`x>=5` or `x<=-1`

Explanation:

We need to consider two ranges since the absolute value will make the function change sign at some value. We first need to find this value, which we do by finding when the bit inside the absolute value is equal to `0`:
`2x-4=0`

`2x=4`

`x=2`

So, we need to look at when `x>2` and when `x<`2

When `x>2`
Since the value inside the absolute value function is positive in this range, we can just remove it:
`2x-4>=6`

`2x-cancel(4+4)>=6+4`

`2x>=10`

`x>=10/2`

`x>=5`

When `x<2`
In this range, the function in the absolute value will be negative, so we multiply it by `-1`:
`-(2x-4)>=6`

`-2x+4>=6`

`-2x+cancel(4-4)>=6-4`

`-2x>=2`

Now we want to divide by `-2`, but we need to be careful since dividing by a negative number flips the sign of the inequality:
`x<=2/-2`

`x<=-1`

Combining the two ranges
We can just combine the two ranges to get that `x>=5` or `x<=-1`