Question

How do you solve the inequality `abs(3-x)<=3`?

Answer 1

`0<=x<=6`

Explanation:

`"Given the inequality " |x|<=a`

`"the solution is always of the form " -a<=x<=a`

`rArr-3<=color(red)(3-x)<=3`

Isolate x in the centre interval while obtaining numeric values in the 2 end intervals.

subtract 3 from ALL intervals.

`-3-3<=cancel(3)cancel(-3)-x<=3-3`

`rArr-6<=-x<=0`

multiply by - 1 to obtain x

`color(blue)"Note"` when multiplying/dividing an inequality by a `color(blue)"negative"` value we must `color(red)" reverse the inequality symbol"`

`rArr6>=x>=0larrcolor(red)" reverse symbol"`

`rArrx<=6color(red)" and " x>=0`

`rArr0<=x<=6" is the solution"`

`x in [0,6]larrcolor(red)" in interval notation"`

Answer 2

`0≤x≤6`, `x\in [0, 6]`

Explanation:

We can rewrite the equation as `-3≤3-x≤3`. Subtract both sides by `3` to get `-6≤-x≤0`. Now, we multiply both sides by `-1` and flip the inequality signs: `6≥x≥0`. This can be rewritten to `0≤x≤6`.

Other notations for this answer: `x\in [0, 6]`