Question

Solve the inequalities write solution in both set and interval notations

Answer 1

`\begin{gathered} SN\colon\lbrace x|x\le-(152)/(13)\rbrace \\ IN\colon(-\infty,-(152)/(13)\rbrack \end{gathered}`

Step-by-step explanation

We want to solve the given inequality:

`(3x)/(4)-(8x)/(2)\ge38`

First, multiply both sides of the inequality by 4:

`\begin{gathered} ((3x)/(4)\cdot4)-((8x)/(2)\cdot4)\ge38\cdot4 \\ \Rightarrow3x-16x\ge152 \end{gathered}`

Now, simplify the left-hand side of the inequality:

`-13x\ge152`

Finally, divide both sides of the inequality by -13. When this is done, the sign changes direction:

`\begin{gathered} (-13)/(-13)x\le(152)/(-13) \\ \Rightarrow x\le-(152)/(13) \end{gathered}`

In set notation, we have:

`\mleft\lbrace x|x\le-(152)/(13)\mright\rbrace`

In interval notation, we have:

`(-\infty,-(152)/(13)\rbrack`