Can someone please help me with this absolute value inequality, can I draw a graph for it? $∣ ∣ ∣ \frac{x + 4}{x + 2} ∣ ∣ ∣ \leq 1$ $|(x+4)/(x+2)|<=1$ ?

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Can someone please help me with this absolute value inequality, can I draw a graph for it? $∣ ∣ ∣ \frac{x + 4}{x + 2} ∣ ∣ ∣ \leq 1$ $|(x+4)/(x+2)|<=1$ ?

1 Answer

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Answer 1 · 2018-03-27T09:31:39

$(- \infty, - 3]$ $(-oo,-3]$

Explanation:

$∣ ∣ ∣ \frac{x + 4}{x + 2} ∣ ∣ ∣ \leq 1$ $|(x+4)/(x+2)|<=1$

We know by the absolute value property that we have to solve both:

$\frac{x + 4}{x + 2} \leq 1$ $(x+4)/(x+2)<=1$ and $- (\frac{x + 4}{x + 2}) \leq 1$ $-((x+4)/(x+2))<=1$

For:

$\frac{x + 4}{x + 2} \leq 1$ $(x+4)/(x+2)<=1$

Subtract $1$ $1$

$\frac{x + 4}{x + 2} - 1 \leq 0$ $(x+4)/(x+2)-1<=0$

Add LHS:

$\frac{(x + 4) - (x + 2)}{x + 2} \leq 0$ $((x+4)-(x+2))/(x+2)<=0$

Simplify:

$\frac{2}{x + 2} \leq 0$ $2/(x+2)<=0$

Divide by 2:

$\frac{1}{x + 2} \leq 0$ $1/(x+2)<=0$

There is no solution for zero, (undefined division by zero)

only $x < - 2$ $x<-2$

For:

$- (\frac{x + 4}{x + 2}) \leq 1$ $-((x+4)/(x+2))<=1$

Subtract 1:

$- \frac{x + 4}{x + 2} - 1 < + 0$ $-(x+4)/(x+2)-1<+0$

Multiply by $- 1$ $-1$ :

$\frac{x + 4}{x + 2} + 1 \leq 0$ $(x+4)/(x+2)+1<=0$

Add LHS:

$\frac{2 x + 6}{x + 2} \leq 0$ $(2x+6)/(x+2)<=0$

Solving for zero:

$\frac{2 x + 6}{x + 2} = 0$ $(2x+6)/(x+2)=0$

$x = - 3$ $x=-3$

We now need to look at :

$∣ ∣ ∣ \frac{x + 4}{x + 2} ∣ ∣ ∣ - 1 \leq 0$ $|(x+4)/(x+2)|-1<=0$

For:

$x < - 3$ $x<-3$

$0 \leq ∣ ∣ ∣ \frac{x + 4}{x + 2} ∣ ∣ ∣ < 1$ $0<=|(x+4)/(x+2)|<1$

So:

$∣ ∣ ∣ \frac{x + 4}{x + 2} ∣ ∣ ∣ - 1 \leq 0$ $|(x+4)/(x+2)|-1<=0$

For $x > - 3$ $x> -3$ , $x \neq - 2$ $x!=-2$

$∣ ∣ ∣ \frac{x + 4}{x + 2} ∣ ∣ ∣ > 1$ $|(x+4)/(x+2)|>1$

So:

$∣ ∣ ∣ \frac{x + 4}{x + 2} ∣ ∣ ∣ - 1 > 0$ $|(x+4)/(x+2)|-1>0$

So only: $x \leq - 3$ $x<=-3$

Solution set:

$(- \infty, - 3]$ $(-oo,-3]$

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Can someone please help me with this absolute value inequality, can I draw a graph for it? $∣ ∣ ∣ \frac{x + 4}{x + 2} ∣ ∣ ∣ \leq 1$ $|(x+4)/(x+2)|<=1$ ?

1 Answer

Explanation:

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Science

Math

Humanities

... and beyond

Can someone please help me with this absolute value inequality, can I draw a graph for it? |(x+4)/(x+2)|<=1∣∣∣x+4x+2∣∣∣≤1|(x+4)/(x+2)|<=1?

1 Answer

Explanation:

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Can someone please help me with this absolute value inequality, can I draw a graph for it? $∣ ∣ ∣ \frac{x + 4}{x + 2} ∣ ∣ ∣ \leq 1$ $|(x+4)/(x+2)|<=1$ ?