How do you solve the inequality $1 + \frac{5}{a - 1} \leq \frac{7}{6}$ $1+5/(a-1)<=7/6$ ?

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Answer 1 · 2018-04-11T17:35:19

$(- \infty, 1) \cup [31, \infty)$ $color(blue)((-oo,1)uu[31,oo)$

$1 + \frac{5}{a - 1} \leq \frac{7}{6}$ $1+5/(a-1)<=7/6$

Subtract $\frac{7}{6}$ $7/6$ form both sides:

$\frac{5}{a - 1} - \frac{1}{6} \leq 0$ $5/(a-1)-1/6<=0$

Add LHS:

$\frac{30 - (a - 1)}{6 (a - 1)} \leq 0$ $(30-(a-1))/(6(a-1))<=0$

$\frac{31 - a}{6 (a - 1)} \leq 0$ $(31-a)/(6(a-1))<=0$

Multiply by $6$ $6$ :

$\frac{31 - a}{a - 1} \leq 0$ $(31-a)/(a-1)<=0$

So we need:

$\frac{+}{-}$ $+/-$ and $\frac{-}{+}$ $-/+$

$31 - a \geq 0 \Rightarrow a \leq 31$ $31-a>=0=>a<=31$

$a - 1 < 0 \Rightarrow a < 1$ $a-1< 0=>a<1$

$31 - a \leq 0 \Rightarrow a \geq 31$ $31-a<=0=>a>=31$

$a - 1 > 0 \Rightarrow a > 1$ $a-1>0=>a>1$

$a < 1$ $a<1$ and $a \geq 31$ $a>=31$

$(- \infty, 1) \cup [31, \infty)$ $color(blue)((-oo,1)uu[31,oo)$

How do you solve the inequality 1+5a−1≤761+5/(a-1)<=7/6?