# How do you solve and graph abs(k-5)<=4?

Apr 8, 2017

$k \in \left[1 , 9\right]$ #### Explanation:

Method 1: Split the absolute value possibilities

{: ("possibility: "(k-5) < 0 (rarr k < 5),color(white)("X")"or"color(white)("X"),"possibility: "(k-5)>=0 (rarr k>=5)), ("then " abs(k-5)<=4,,"then " abs(k-5) <=4), (rarr 5-k <=4,,rarrk-5 <-4), (color(white)("XX")1 <= k,,color(white)("XX")k <=9), (k < 5" and "k>=1,color(white)("X")" or "color(white)("X"),k>=5" and "k<=9) :}
Combining the two possibilities we have
$\textcolor{w h i t e}{\text{XXX}} 1 \le k \le 9$

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Method 2: square the two sides then check critical and sample interval points

Given $\left\mid k - 5 \right\mid \le 4$

Squaring both sides:
$\textcolor{w h i t e}{\text{XXX}} {k}^{2} - 1 o k + 25 \le 16$

$\textcolor{w h i t e}{\text{XXX}} k - 10 k + 9 \le 0$

$\textcolor{w h i t e}{\text{XXX}} \left(k - 1\right) \left(k - 9\right) \le 0$

Giving the critical points $\textcolor{red}{k = 1}$ and $\textcolor{red}{k = 9}$
We can then select arbitrary interval points:
for $k < 1$, $\textcolor{w h i t e}{\text{XX}}$I selected $\textcolor{b l u e}{k = 0}$
for $k \in \left(1 , 9\right)$, I selected $\textcolor{b l u e}{k = 5}$
for $k > 9$, $\textcolor{w h i t e}{\text{XX}}$I selected $\textcolor{b l u e}{k = 10}$

{: (," | ",color(blue)(k=0),color(red)(k=1),color(blue)(k=5),color(red)(k=9),color(blue)(k=10)), (abs(k-5)," | ",5,4,0,4,5), ("? ? "<=4," | ",F,T,T,T,F) :}

So the inequality is true for $k = 1$, points between $k = 1$ and $k = 9$, and for $k = 9$

That is the inequality is valid for $k \in \left[1 , 9\right]$

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Plotting is simply a matter of drawing this result on the number line.