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