Question

Solve the inequalities write solution in both set and interval notations

Answer 1

ANSWER

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

EXPLANATION

We want to solve the given inequality:

$\frac{3x}{4}-\frac{8x}{2}\ge38$

First, multiply both sides of the inequality by 4:

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

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:

$\begin{gathered} \frac{-13}{-13}x\le\frac{152}{-13} \\ \Rightarrow x\le-\frac{152}{13} \end{gathered}$

In set notation, we have:

$\mleft\lbrace x|x\le-\frac{152}{13}\mright\rbrace$

In interval notation, we have:

$(-\infty,-\frac{152}{13}\rbrack$