Q9. - Algebra 2

Question

Graph each inequality. 

 

$y\le 3\left|x\right|-1$

Step-by-Step Solution

Answer

The graph of the inequality $y\le 3\left|x\right|-1$ is provided below.

Step-1 – Apply the concept of graphing the inequality

The steps to graph the inequality are provided below.

1. If the inequality contains greater than or less than sign then the boundary of the line will be dashed. If the inequality contains signs of greater than or equal to or less than or equal to then the boundary of the line will be solid. 

2. Select a point (known as test point) from the plane that does not lie on the boundary on the line and substitute it in the inequality. 

3. If the inequality is true then shade the region that contains the test point otherwise shade the other region when inequality is false.

Step-2 – Interpret the sign of the inequality

Consider the inequality provided below.

$y\le 3\left|x\right|-1$

The absolute value of a function is expressed as, if x is a real number then absolute value of x is defined as.

$\left|x\right|=x$ when x is greater than or equal to 0 . In other words, the absolute value of x is x when x is either positive or zero.

$\left|x\right|=-x$ when x is less than 0 . In other words, the absolute value of x is opposite of x when x is negative.

The inequality is split as $y\le 3x-1\text{ and }y\ge -3x-1$.

The inequality contains the sign of less than or equal to and greater than or equal to respectively.

Therefore, the boundary lines will be dashed.

Step-3 – Apply the concept of slope-intercept form

Equation of line in slope intercept form is expressed below.

$y=mx+c$

Where m is the slope and c is the intercept of y-axis.

Step-4 –Write the equations in slope-intercept form

Consider the equation $y=3x-1$. 

Rewrite the equation in form of slope-intercept form.

$y=3x+\left(-1\right)$

Now, the equation is in the form $y=mx+c$. Here slope m of the line is 3 and intercept of y-axis c is $-1$.

Consider the equation $y=-3x-1$. 

Rewrite the equation in form of slope-intercept form.

$y=-3x+\left(-1\right)$

Now, the equation is in the form $y=mx+c$. Here slope m of the line is $-3$ and intercept of y-axis c is $-1$.

Step-5 – Graph the inequality

The corresponding equation is $y=3x-1$.

Take a test point that does not lie on the boundary of the line, say $\left(1,0\right)$. 

Substitute the point $\left(1,0\right)$ in the inequality and check whether it’s true or not.

$\begin{array}{c}0\le 3\left(1\right)-1\\ 0\le 2\end{array}$

This is true.

Therefore, shade the region containing the point $\left(1,0\right)$.

The corresponding equation is $y=3x-1$.

Take a test point that does not lie on the boundary of the line, say $\left(1,0\right)$. 

Substitute the point $\left(1,0\right)$ in the inequality and check whether it’s true or not.

$\begin{array}{c}0\ge -3\left(1\right)-1\\ 0\ge -4\end{array}$

This is true.

Therefore, shade the region containing the point $\left(1,0\right)$.

Thus, the common shaded region is provided below.

Thus, the graph of the inequality is shown above.