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# Algebra

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

Aug 9, 2024

#1
+1790
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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

Aug 9, 2024
edited by NotThatSmart  Aug 9, 2024
edited by NotThatSmart  Aug 9, 2024
edited by NotThatSmart  Aug 9, 2024
edited by NotThatSmart  Aug 9, 2024
edited by NotThatSmart  Aug 9, 2024
edited by NotThatSmart  Aug 9, 2024
edited by NotThatSmart  Aug 9, 2024
edited by NotThatSmart  Aug 9, 2024