Question

In Exercises 15-20. Sketch the region given by the set.

(a) $\left\{\left(x,y|y<3\right)\right\}$

(b) $\left\{\left(x,y|x=-4\right)\right\}$

Short answer

(a) The region given by the set $\left\{\left(x,y\right)\left|y<3\right.\right\}$is provided below.

(b) The region given by the set $\left\{\left(x,y\right)\left|x=-4\right.\right\}$ is provided below.

Step-by-step solution

a.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.

$y<3$

The inequality contains the sign of less than.

Therefore, the boundary line will be dashed.

Step 3. Graph the inequality

Graph the inequalities $y<3$.

Draw the line $y=3$.

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

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

$0<3$

This is true.

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

Thus, the shaded region is provided below.

Graphical interpretation

The region provided by the set $\left\{\left(x,y\right)\left|y<3\right.\right\}$ is the shaded region and consists all the points whose y-coordinate is less than 3.

b.Step 1. 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 2. Write the equations in slope-intercept form

Consider the equation $x=-4$.

This is a vertical line. The slope of vertical lines are undefined. Therefore, this equation cannot be put to slope intercept form.

Step 3. Construct a table of values

Consider real values for x and corresponding to it find the values of y.

Step 3. Graph the equation

Plot the ordered pairs on the Cartesian plane to obtain the graph of the equation $x=-4$.

Graphical interpretation

The region provided by the set$\left\{\left(x,y\right)\left|x=-4\right.\right\}$ is the dark red line. All the points that lie in the region have x-coordinate as$-4$ for all y-coordinates. The line is a vertical line four units left of the origin.