Question

x-(6)/x<=-1

Answer 1

Is this supposed to be:

x - [ (6 )/ x] <= - 1

Answer 2

check the pic @Directrix

Answer 3

start with
\[x-\frac{6}{x}+1\leq 0\] then add

Answer 4

\[x-\frac{ 6 }{ x }\le -1\]
\[x \neq 0 ,\because~then~\frac{ 6 }{ x }~is ~\not~defined.\]
case 1.
let x>0
then \[x^2-6\le -x (multiply~ by~ x )\]
\[x^2+x-6 \le 0\]
\[x^2+x+\left( \frac{ 1 }{ 2 } \right)^2-\left( \frac{ 1 }{ 2 } \right)^2-6\le 0\]
\[\left( x+\frac{ 1 }{ 2 } \right)^2 \le \frac{ 1 }{ 4 }+6\]
\[\left( x+\frac{ 1 }{ 2 } \right)^2\le \left( \frac{ 5 }{ 2 } \right)^2\]
\[\left| x+\frac{ 1 }{ 2 } \right|\le \frac{ 5 }{ 2 }\]
\[-\frac{ 5 }{ 2 }\le x+\frac{ 1 }{ 2 }\le \frac{ 5 }{ 2 }\]
subtract 1/2 from each
\[-\frac{ 5 }{ 2 }-\frac{ 1 }{ 2 }\le x \le \frac{ 5 }{ 2 }-\frac{ 1 }{ 2 }\]
\[-3 \le x \le 2\]
but x>0
hence \[0< x \le 2\]
case 2.
let x<0
\[x^2-6 \ge -x (change~ inequality )\]
\[x^2+x -6 \ge 0\]
solve again and find x

Answer 5

Let's solve your inequality step-by-step. x6x1 x21 Let's find the critical points of the inequality. x26x=1 x2x(Multiply both sides by x) x2x)=x)(Subtract -x from both sides) x2+x6=0 (x2)(x+3)=0(Factor left side of equation) x2=0 or x+3=0(Set factors equal to 0) x=2 or x=3 ________________________________________ Check possible critical points. x=2(Works in original equation) x=3(Works in original equation) ________________________________________ Critical points: x=2 or x=3(Makes both sides equal) x=0(Makes left denominator equal to 0) Check intervals in between critical points. (Test values in the intervals to see if they work.) x3(Works in original inequality) x<0(Doesn't work in original inequality) 0<x2(Works in original inequality) x 2(Doesn't work in original inequality)

Answer 6

we cannot multiply by x as long as we don't mention x>0 or x<0