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Find all real numbers $t$ such that $\frac{2}{3} t - 1 < t + 7 \le -2t + 15$. Give your answer as an interval.

Guest Dec 10, 2017

#2
+5888
+1

$$\frac{2}{3}t-1\,\text{<}\,t+7\le-2t+15 \\~\\ \text{Let's split it into two inequalities.} \\~\\ \frac{2}{3}t-1\,\text{<}\,t+7\\~\\ -1-7\,\text{<}\,t-\frac{2}{3}t\\~\\ -8\,\text{<}\,\frac{1}{3}t\\~\\ -24\,\text{<}\,t\\~\\ \text{and}\\~\\ t+7\le-2t+15\\~\\ t+2t\le15-7\\~\\ 3t\le8\\~\\ t\le\frac{8}{3}\\~\\ \text{So t is greater than 24 but less than or equal to 8/3.}\\~\\ \text{t is in the interval:}\quad (-24,\frac{8}{3}]$$

hectictar  Dec 10, 2017
edited by hectictar  Dec 10, 2017
edited by hectictar  Dec 10, 2017
edited by hectictar  Dec 10, 2017
edited by hectictar  Dec 10, 2017
edited by hectictar  Dec 10, 2017
edited by hectictar  Dec 10, 2017
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#1
+11
0

Interval or integral?

GLTiffin  Dec 10, 2017
#2
+5888
+1

$$\frac{2}{3}t-1\,\text{<}\,t+7\le-2t+15 \\~\\ \text{Let's split it into two inequalities.} \\~\\ \frac{2}{3}t-1\,\text{<}\,t+7\\~\\ -1-7\,\text{<}\,t-\frac{2}{3}t\\~\\ -8\,\text{<}\,\frac{1}{3}t\\~\\ -24\,\text{<}\,t\\~\\ \text{and}\\~\\ t+7\le-2t+15\\~\\ t+2t\le15-7\\~\\ 3t\le8\\~\\ t\le\frac{8}{3}\\~\\ \text{So t is greater than 24 but less than or equal to 8/3.}\\~\\ \text{t is in the interval:}\quad (-24,\frac{8}{3}]$$

hectictar  Dec 10, 2017
edited by hectictar  Dec 10, 2017
edited by hectictar  Dec 10, 2017
edited by hectictar  Dec 10, 2017
edited by hectictar  Dec 10, 2017
edited by hectictar  Dec 10, 2017
edited by hectictar  Dec 10, 2017
#3
+80874
+2

(2/3)t - 1 < t+ 7 ≤ -2t + 15

First, we have

(2/3) t - 1 < t + 7     multiply through by 3

2t - 3 < 3t + 21    subtract  21, 2t from both sides

-24 < t

Then, we have

t + 7 ≤  -2t+ 15     add 2t to both sides, subtract 7 from both sides

3t ≤ 8    divide both sides by 3

t ≤  8/3

(- 24, 8/3 ]

CPhill  Dec 10, 2017

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