Find all $x$ such that $1-3x < x + 8 \le 7-x$. Give your answer in interval notation.

Rangcr897 Aug 9, 2024

#1**+1 **

Let's split the inequality into two different inequalities and find the overlapping numbers.

Now, let's find an interval for x from both inequalities and find the overlapping zone.

Alright. So we have the two inequalities

\(1-3x < x+8\space\space\space\space\space(1)\\ x + 8 \le 7-x\space\space\space\space\space\space(2)\)

From the first inequality, we have

\(1-3x -7/4\)

From the second inequality, we have

\(x+8 \leq 7-x\\ 2x \leq -1\\ x \leq -1/2\)

Thus, we have the intervals

\(x>-\frac{7}{4}\quad \mathrm{and}\quad \:x\le \:-\frac{1}{2}\)

In interval notation, this is \((-\frac{7}{4},\:-\frac{1}{2}]\)

Thanks! :)

*1500 points

NotThatSmart Aug 9, 2024

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edited by NotThatSmart Aug 9, 2024

edited by NotThatSmart Aug 9, 2024