c To graph the inequality, start by identifying the boundary line.
D
d To graph the inequality, start by identifying the boundary curve.
A
aDiagram:
B
bDiagram:
C
cDiagram:
D
dDiagram:
Practice makes perfect
a To graph the given inequality, we should first find its boundary line. To do so, let's begin by rewriting the inequality into a more familiar format.
Now we can find the boundary line. We will do that by temporarily considering the related equation instead. We find a related equation by changing the inequality sign to an equals sign.
Inequality:& y>3x-3
Equality:& y=3x-3
The equation is written in slope-intercept form, with a slope of 3 and a y-intercept of (0,-3). With this information, we can plot the graph by first marking the line's y-intercept and using the slope to find a second point on the line.
Notice that the inequality is strict. This means the boundary line is not part of the solution set which is why we dash the line. Finally, we must shade the correct side of the line. To do that, we will test a point that does not fall on the line in the original inequality. In this case, the easiest point we can choose is the origin.
Next, we will consider the related equation which will allow us to write the boundary line.
Inequality:& y<3
Equality:& y=3
The equation y=3 is a horizontal line through (0,3) on the y-axis. Like in Part A, we have a strict inequality which means we will dash the boundary line.
Notice that y<3 tells us that every y-value that is less than 3 is part of the solution set. Therefore, we know that we must shade the region below the boundary line.
c Like in Part A, we will start by isolating the y-variable in the inequality.
Now we can find the boundary line. We will do that by temporarily considering the related equation instead. We find a related equation by changing the inequality sign to an equals sign.
Inequality:& y>3/2x-3 [0.5em]
Equality:& y=3/2x-3
The equation is written in slope-intercept form, with a slope of 32 and a y-intercept of (0,-3). With this information, we can plot the graph by first marking the line's y-intercept and using the slope to find a second point on the line.
Notice that the inequality is non-strict which is why we keep the boundary line solid. Finally, we must shade the correct side of the boundary line. To do that, we will test a point in the original inequality that is not on the boundary line. In this case, the easiest point we can choose is the origin.
To graph the boundary line, we will consider the related equation.
Inequality:& y≥x^2-9
Equality:& y=x^2-9
Notice that y=x^2-9 is a vertical translation of the parent function y=x^2 in the negative direction by 9 units.
Like in Part C, we have a non-strict inequality which means the boundary curve is part of the solution set. Therefore, we keep the curve solid. To shade the correct side of the boundary curve, we will substitute a point that is not on the curve into the original inequality. We will choose the origin.
