Q19. - Algebra 2

Question

Graph each equation or inequality

$\text{\hspace{0.17em}\hspace{0.17em} y < 4}\left|x-1\right|$

Step-by-Step Solution

Answer

The graph of the inequality $\text{y < 4}\left|x-1\right|$ is

Step-1 &ndash; Apply the concept of Linear Inequality and absolute value function

A linear inequality resembles a linear equation but with an inequality symbol instead of an equals symbol.

For example, $y\le x+5$ is a linear inequality with y=x+5 as a related linear equation.

Also, the graph of y=x+5 separates the coordinate plane into two region. The line of the this graph acts as a boundary of the two separated region. 

The absolute value function, written as $f\left(x\right)=\left|x\right|$ is defined as follows:

$f\left(x\right)=\left\{\begin{array}{c}-x\text{ if }x<0\\ x\text{ if }x\ge 0\end{array}\right.$

For examples, $\left|-2\right|=2;\left|2\right|=2;\left|3.5\right|=3.5;\left|-3.5\right|=3.5$

Step-2 &ndash; Calculate the boundary

The  given linear equality is $y<4\left|x-1\right|$ and thus the boundary which separates the coordinate plane into two region is $y=4\left|x-1\right|$. As the inequality symbol is $<$ and therefore the boundary line will a dotted line which means that the line is not included.

Step-3 &ndash; Graph the boundary line

Find the value of y corresponding to the value of x.

x	$\left|x-1\right|$	$y=4\left|x-1\right|$
$-2$	$\begin{array}{c}\left|-2-1\right|\\ =\left|-3\right|\\ =3\end{array}$	$\begin{array}{l}4\left(3\right)\\ =12\end{array}$
$-1$	$\begin{array}{c}\left|-1-1\right|\\ =\left|-2\right|\\ =2\end{array}$	$\begin{array}{l}4\left(2\right)\\ =8\end{array}$
0	$\begin{array}{c}\left|0-1\right|\\ =\left|-1\right|\\ =1\end{array}$	$\begin{array}{l}4\left(1\right)\\ =4\end{array}$
1	$\begin{array}{c}\left|1-1\right|\\ =\left|0\right|\\ =0\end{array}$	$\begin{array}{l}4\left(0\right)\\ =0\end{array}$
2	$\begin{array}{c}\left|2-1\right|\\ =\left|1\right|\\ =1\end{array}$	$\begin{array}{l}4\left(1\right)\\ =4\end{array}$
3	$\begin{array}{c}\left|3-1\right|\\ =\left|2\right|\\ =2\end{array}$	$\begin{array}{l}4\left(2\right)\\ =8\end{array}$

Hence by using the values in the table, the graph is obtained as shown below:

Step-4 &ndash; Test a point on inequality

Choose a point which is not on a boundary line. 

Since the point (0, 0) is not on a boundary line, test the point (0, 0) on the inequality $y<4\left|x-1\right|$.

Therefore, 

$\begin{array}{l}0<4\left|0-1\right|\\ 0<4\left|-1\right|\\ 0<4\left(1\right)\\ 0<4\end{array}$

which is true.

Therefore, shade the region which contain (0, 0).

Hence the graph of the linear inequality is as shown below: