Graphing Linear Inequalities

There are two major steps to writing an inequality when given its graph.

1. Write an equation for the boundary line.
2. Determine the inequality symbol and complete the inequality.

Let's get started by focusing on the boundary line.

Writing the Boundary Line Equation

It only takes two points to create a unique equation for any line, so let's identify two points on the boundary line.

Here we have identified two points, (0, 3) and (3,4), and indicated the horizontal and vertical changes between them. This gives us the rise and run of the graph, which will give us the slope m. rise/run=1/3 ⇔ m= 1/3 One of the points we selected, (0, 3), is also the y-intercept, which is great news! This means we can combine the slope m and the y-intercept at the point (0, b) to write an equation for the boundary line in slope-intercept form. y= mx+ b ⇒ y= 1/3x+ 3

Forming the Inequality

To finish forming the inequality, we need to determine the inequality symbol. This means replacing the equals sign with a blank space since it is still unknown. y ? 1/3x+3 To figure out what the symbol should be, we need a test point that lies within the solution set.

We will substitute ( 0, 0) for this test, then use the inequality symbol that suits the resulting statement.

Zero is less than 3. We can infer that, of the four inequality symbols, only two would make this a true statement, < or ≤. Returning to the given graph one last time, we can see that the boundary line is dashed, not solid. This implies that the inequality is strict. We can now form our final inequality. y< 1/3x+3 The inequality can also be written in standard form.

To graph an inequality, we should start by drawing its boundary line. We can write the equation of this line by replacing the inequality sign with an equals sign. ccc Inequality & & Boundary Line [0.8em] y> 3x+2 & & y= 3x+2 Since the line is already written in slope-intercept form, we can use its y-intercept, 2, and its slope, 3, to draw its graph. Note that we have a strict inequality, so the boundary line will be dashed.

We will use a test point to determine whether we have to shade above or below the line. If substituting this point in the inequality produces a true statement, we shade the region that contains the point. Otherwise, we shade the opposite region. For simplicity, we will use ( 0, 0) as a test point.

Since the test point produced a false statement, we will shade the region that does not contain (0,0).

We have shaded above the boundary line. Since the boundary line is dashed, the area of the solution set is above the boundary line and does not include the line. Therefore, the answer is C.

Let's start by graphing the given inequality. To do so, we must first identify the boundary line. To write the equation of the boundary line, we replace the inequality sign with an equals sign. ccc Inequality & & Boundary Line [0.4em] y ≤ 1/4x-1 & & y = 1/4x+( -1) Since the line is already written in slope-intercept form, we will use its slope 14 and its y-intercept -1 to graph it. Note that the inequality is non-strict, so the line will be solid.

Now that we have graphed the boundary line, we will determine the region to shade. To do so, we will test a point. If substituting the point into the inequality produces a true statement, we will shade the region that contains the point. Otherwise, we will shade the opposite region. For simplicity, we will test the point ( 0, 0).

Since the test point produced a false statement, we will shade the region that does not contain (0,0).

Note that the solution set contains all the points that are below the line, including the boundary line. Therefore, the region of the coordinate plane not included in the graph is the region above the line, not including the line. This can be expressed by reversing the inequality sign and making the inequality strict. y> 1/4x-1 Let's see how the graph of this inequality looks like.

First, we will write an expression for Heichi's purchases. To do so, we can assign variables to the number of pizzas and sodas that Heichi can buy. We can call the number of pizzas he buys p and the number of bottles of sodas s. The total price of his purchases can be written as an expression. 6p + 2s Since Heichi can spend at most $30, we know that the total purchases have to be less than or equal to 30. 6p + 2s ≤ 30 We can see that this is a linear inequality. We might have an easier time graphing the inequality by writing it in slope-intercept form first.

We can see that the slope is - 13, while the y-intercept is 5. We can use this information to graph the boundary line of the inequality.

Note that we should only consider the first quadrant because a negative number of sodas or pizzas does not make sense in this context.

Finally, we will use a test point to know which region should be shaded. For simplicity, we will substitute ( 0, 0) into our inequality.

We can see that substituting the point resulted in a true statement. Therefore, we will shade the region below the boundary line.

This corresponds to option A.