Question

What is the solution to the inequality?

Express yourself in interval notation.​

Answer 1

`\begin{bmatrix}\mathrm{Solution:}\:&\:x\ge (20)/(23)\:\\ \:\mathrm{Decimal:}&\:x\ge \:0.86956\dots \\ \:\mathrm{Interval\:Notation:}&\:[(20)/(23),\:\infty \:)\end{bmatrix}`

Explanation:

`-(2)/(3)\left(2x-(1)/(2)\right)\le (1)/(5)x-1`

Expand
`-(2)/(3)\left(2x-(1)/(2)\right)\le (1)/(5)x-1`
`-(4)/(3)x+(1)/(3)`

`-(4)/(3)x+(1)/(3)\le (1)/(5)x-1`

`\mathrm{Subtract\:}(1)/(3)\mathrm{\:from\:both\:sides}`

`-(4)/(3)x+(1)/(3)-(1)/(3)\le (1)/(5)x-1-(1)/(3)`

`Simplify`

`-(4)/(3)x\le \:-(4)/(3)+(1)/(5)x`

`\mathrm{Subtract\:}(1)/(5)x\mathrm{\:from\:both\:sides}`

`-(4)/(3)x-(1)/(5)x\le \:-(4)/(3)+(1)/(5)x-(1)/(5)x`

`Simplify`

`-(23)/(15)x\le \:-(4)/(3)`

`\mathrm{Multiply\:both\:sides\:by\:-1\:\left(reverse\:the\:inequality\right)}`

`\left(-(23)/(15)x\right)\left(-1\right)\ge \left(-(4)/(3)\right)\left(-1\right)`

`\mathrm{Simplify}`

`(23)/(15)x\ge (4)/(3)`

`\mathrm{Multiply\:both\:sides\:by\:}15`

`15\cdot (23)/(15)x\ge (4\cdot \:15)/(3)`

`Simplify`

`23x\ge \:20`

`\mathrm{Divide\:both\:sides\:by\:}23`

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

`\mathrm{Simplify}`

Hence the final answer is
`\begin{bmatrix}\mathrm{Solution:}\:&\:x\ge (20)/(23)\:\\ \:\mathrm{Decimal:}&\:x\ge \:0.86956\dots \\ \:\mathrm{Interval\:Notation:}&\:[(20)/(23),\:\infty \:)\end{bmatrix}`