Graph the inequality.
y is less than negative one fourth times x minus 3
There are different ways to graph an inequality, but one common method is to graph the boundary line for the related equation (with an equal sign instead of an inequality sign) and then shade the side that satisfies the inequality.
To graph the equation y = -1/4 x - 3, we can use the slope-intercept form, which means that the y-intercept is -3 and the slope is -1/4. This means that if we move 4 units to the right (increasing x by 4), we would move 1 unit down (decreasing y by 1). We can plot two points on the line using these values (or any other values that satisfy the equation), such as:
- When x = 0, y = -3, so (0, -3) is one point on the line.
- When x = 4, y = -4, so (4, -4) is another point on the line.
We can then draw a straight line through these two points.
To graph the inequality y < -1/4 x - 3, we need to shade the region below the line (because y is less than, not equal to, the value on the line). We can test any point that is not on the line to see whether it satisfies the inequality and lies below the line. For example, the point (0, 0) is a convenient test point that lies above the line. If we substitute x = 0 and y = 0 into the inequality, we get:
0 < -1/4 (0) - 3
0 < -3
This statement is false, so (0, 0) is not in the shaded region. We can therefore shade the opposite region, below the line:
(Note: the shading is not always easy to see on a computer, but it should be clear that the region below the line is shaded.)
To graph the inequality "y is less than negative one-fourth times x minus 3," we can follow these steps:
1. Start by graphing the equation y = -1/4x - 3 as a solid line. This line represents the boundary between the solutions that are less than the inequality and the solutions that are not.
2. To graph y < -1/4x - 3, we need to shade the region below the boundary line because we want the solutions that are less than the line.
Here is a step-by-step guide on how to graph this inequality:
Step 1: Draw the x and y coordinate axes on your graph paper.
Step 2: Plot the y-intercept. The equation is in slope-intercept form (y = mx + b), where the y-intercept is -3. So, plot the point (0, -3) on the y-axis.
Step 3: Find another point on the line. To do this, we can use the slope (m) of the line. The coefficient of x in the equation is -1/4, which means the slope is -1/4. This tells us that as we move 4 units to the right, we move 1 unit downward. So, starting from the y-intercept point (0, -3), move 4 units to the right (one step), and 1 unit downward (since slope is negative), and plot the resulting point as (4, -4.25).
Step 4: Now, draw a straight line connecting the two plotted points (0, -3) and (4, -4.25). This is the boundary line y = -1/4x - 3.
Step 5: Since the inequality is y < -1/4x - 3, shade the region below the boundary line to represent the solutions that are less than the line.
The final graph should have a solid line y = -1/4x - 3, with the region below it shaded.