Which shows the graph of the solution set of 2x + y < 4?

Answer and step-by-step explanation:

This inequality can be written similarly to an equation in slope-intercept form, y=mx+b, by isolating the y.  Subtract 2x from each side:

2x+y-2x < 4-2x

y < 4-2x

y < -2x+4

This makes the related equation y = -2x+4.  In this equation, the slope is -2 and the y-intercept is 4.  This means we first plot the point (0, 4) on the y-axis.  The slope is -2, which can be written as -2/1; this means to find the next point, we go from down 2 and right 1 from the y-intercept to plot the next point (1, 2); then down 2 and right 1 for the next point (2, 0).  Since this is strictly less than, and not less than or equal to, we use a dashed line instead of a solid line.

Since this is "less than", this means that we shade the graph under the line.

we have

Using a graphing tool

see the attached figure

the solution is the shaded area below the line

The answer in the attached figure

the range of a function is the range of y-values for which it is defined, or exists. given that we are only looking at the y-values, we can already discount the first two answers as they deal with the domain of the function (range of x-values).

now, what we need to do is find the smallest and highest y-values for which the function is defined. if you'd like, you could take a ruler and start at the very bottom of the graph, find the lowest point, then move the ruler higher and higher until you reach the highest point. doing so either in your head or with an actual ruler, we can see that the lowest y-value is -5 and the highest y-value is -1.

the question now remains as to whether they are included in the range (fourth answer) or excluded (third answer). the key here is to recognise whether they are marked with an open or a closed circle. since both (-2, -5) and (2, -1) are marked with a closed circle, they are points that exist on the graph and are thus included in the range (note that if they were marked with an open circle like (4, -2) then that would signify that there is a break in the graph at that point).

thus, the fourth answer, (y | -5 ≤ y ≤ -1), is correct.

also note that usually it is good practice to scan your eye inbetween the highest and lowest point for any breaks in the graph - we do have one here at (4, -2), however -2 is still included in the range because there is another point that is included in the graph with the same y-value, ie. point (1, -2).

i hope that but if you require any further explanation don't hesitate to comment below.