Graphing Absolute Value Inequalities

We will follow three steps to graph the given absolute value inequality.

1. Draw the boundary line.
2. Test a point that is not on the boundary line.
3. Shade the solution set.

Drawing the Boundary Line

The boundary line of an inequality can be determined by replacing the inequality symbol with an equal sign. Inequality & Boundary Line y ≥ |3x-12| & y = |3x-12| Notice that the equation of the boundary line is an absolute value equation. Let's rewrite it in vertex form to find its vertex. y = |3x-12| ⇓ y = 3|x- 4|+ 0 Its vertex is ( 4, 0). The graph of the boundary line should be symmetric about the line x=4. To graph the boundary line, we can make a table of values using some x-values greater and less than 4.

We now plot the points and draw the boundary line. Since the inequality is non-strict, the boundary line will be solid.

Testing a Point

In order to decide which part of the plane to shade, we can test a point which is not on the boundary line. Let's test the point ( 0, 0).

If the point satisfies the inequality, we shade the region that contains the point. Otherwise, we shade the region that does not contain the point. Let's substitute the test point into the given inequality.

Since the point does not satisfy the inequality, it is not a solution to the inequality.

Shading the Solution Set

We will shade the region that does not contain the point because the test point is not a solution.

We can see that Graph A is the same as the graph we drew.

We will follow three steps to graph the given absolute value inequality.

1. Draw the boundary line.
2. Test a point that is not on the boundary line.
3. Shade the solution set.

Drawing the Boundary Line

The boundary line of an inequality can be determined by replacing the inequality symbol with an equal sign. Inequality & Boundary Line y < |0.75x|+2 & y = |0.75x|+2 Notice that the equation of the boundary line is an absolute value equation. Let's rewrite it in vertex form to find its vertex. y = |0.75x|+2 ⇓ y = 0.75|x- 0|+ 2 Its vertex is ( 0, 2). The graph of the boundary line should be symmetric about the y-axis. To graph the boundary line, we can make a table of values using some x-values below and above 0.

We now plot the points and draw the boundary line. Since the inequality is strict, the boundary line will be dashed.

Testing a Point

In order to decide which part of the plane to shade, we can test a point which is not on the boundary line. Let's test the point ( 0, 0).

If the point satisfies the inequality, we shade the region that contains the point. Otherwise, we shade the region that does not contain the point. Let's substitute the test point into the given inequality.

Since the point satisfies the inequality, it is a solution to the inequality.

Shading the Solution Set

We will shade the region that contains the point because the test point is a solution.

This graph matches with the graph in C.

We will follow three steps to graph the given absolute value inequality.

1. Draw the boundary line.
2. Test a point that is not on the boundary line.
3. Shade the solution set.

Drawing the Boundary Line

In this case, we will first isolate the y-variable.

The boundary line of an inequality can be determined by replacing the inequality symbol with an equal sign. Inequality & Boundary Line y > 5 |1/4x+2|-10 & y = 5 |1/4x+2|-10 Notice that the equation of the boundary line is an absolute value equation. Let's rewrite it in vertex form to find its vertex.

We can now identify the vertex. y = 5/4|x+8|-10 ⇓ y = 5/4|x-( - 8)|+ ( - 10) Its vertex is ( - 8, - 10). Therefore, the graph of the boundary line should be symmetric about the line x=- 8. To graph the boundary line, we can make a table of values using some x-values below and above - 8.

We now plot the points and draw the boundary line. Since the inequality is strict, the boundary line will be dashed.

Testing a Point

In order to decide which part of the plane to shade, we can test a point which is not on the boundary line. Let's test the point ( - 8, 0).

If the point satisfies the inequality, we shade the region that contains the point. Otherwise, we shade the region that does not contain the point. Let's substitute the test point into the given inequality.

Since the point satisfies the inequality, it is a solution to the inequality.

Shading the Solution Set

We will shade the region that contains the point because the test point is a solution.

This graph matches with the graph in option B.

To write an inequality for the graph, we will follow two steps.

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

Writing the Boundary Line Equation

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

We see that the vertex of the boundary line is (- 4,0). Recall the vertex form of an absolute value equation whose vertex is at (h,k). y = a|x-h|+k Since the vertex of the boundary line is ( - 4, 0), the corresponding values can be substituted and the expression simplified.

We can substitute a point on the boundary line, different than the vertex, to find the value of a. We know that (- 6,2) lies on the boundary line. Let's substitute these values into our equation and solve for a!

We can now write the equation of the boundary line. y = 1|x+4| ⇔ y = |x+4|

Forming the Inequality

Now that we have the equation of the boundary line, we will determine the inequality symbol.

Since the boundary line is dashed, the inequality will be strict, meaning that it is either < or >. The shaded region is below the boundary line. That is, the y-values are less than |x+4|. Therefore, the inequality symbol should be <. Boundary Line: & y = |x+4| Inequality:& y < |x+4|

We can check our answer by substituting a point in the shaded region into our function and simplifying. If the solution is true, we know our answer is correct. Let's try (- 3,- 2).

We can conclude that the solution set of the inequality is the given graph.

To write an inequality for the graph, we will follow two steps.

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

Writing the Boundary Line Equation

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

We see that the vertex of the boundary line is (5,- 1). Recall the vertex form of an absolute value equation whose vertex is at (h,k). y = a|x-h|+k Since the vertex of the boundary line is ( 5, - 1), the corresponding values can be substituted.

We can substitute a point on the boundary line to find the value of a. Let's use (0,- 5).

We can now write the equation of the boundary line. y = - 0.8|x-5|-1

Forming the Inequality

Now that we have the equation of the boundary line, we will determine the inequality symbol.

Since the boundary line is solid, the inequality will be non-strict, meaning that it is either ≤ or ≥. The shaded region is above the boundary line. That is, the y-values are greater than or equal to |x+4|. Therefore, the inequality symbol should be ≥. Boundary Line: & y = - 0.8|x-5|-1 Inequality:& y ≥ - 0.8|x-5|-1

We can check our answer by substituting a point in the shaded region. If the inequality is correct, then it will produce a true statement. Let's try (0,0).

We can conclude that the solution set of the inequality is the given graph.