Question

1. What is the solution to the inequality?

- 2/3(2x - 1/2) ≤ 1/5 x-1 Express your answer in interval notation.

Answer 1

we are given

`-(2)/(3) (2x-(1)/(2)) \leq(1)/(5)x-1`

Since, we have to solve this inequality

so, we will isolate x on anyone side

step-1:

Multiply both sides by 3/2

`(3)/(2)*-(2)/(3)(2x-(1)/(2))\leq(3)/(2)*((1)/(5)x-1)`

`-(2x-(1)/(2))\leq(3)/(2)*(1)/(5)x-(3)/(2)* 1`

`-(2x-(1)/(2)) \leq(3)/(10)x-(3)/(2)`

step-2:

Distribute negative sign

`-2x+(1)/(2) \leq(3)/(10)x-(3)/(2)`

step-3:

Subtract both sides by 1/2

`-2x+(1)/(2)-(1)/(2) \leq(3)/(10)x-(3)/(2)-(1)/(2)`

`-2x \leq(3)/(10)x-2`

step-4:

Subtract both sides by (3/10)x

`-2x-(3)/(10)x \leq(3)/(10)x-2-(3)/(10)x`

`-2x-(3)/(10)x \leq-2`

`-(23)/(10)x \leq-2`

step-5:

Multiply by sides by -10/23

`-(10)/(23)* -(23)/(10)x \geq-(10)/(23)*-2`

now, we can simplify it

`x \geq(20)/(23)`

now, we can write in interval notation as

`[(20)/(23),\:\infty \:)`................Answer