Graphing Inequalities

Inequalities can be represented graphically using number lines or coordinate planes. Graphing inequalities helps to visualise the solution set and provides a better understanding of the relationships between the variables.

Number Line Representation

When representing a simple inequality on a number line, follow these steps:

1. Draw a number line with the variable’s possible values.

2. Identify the critical value (the value at which the inequality sign changes direction).

3. For strict inequalities (<, >), draw an open circle at the critical value, which indicates that the value is not included in the solution set. For inclusive inequalities (≤, ≥), draw a closed circle at the critical value to show that it is part of the solution set.

4. Shade the appropriate region of the number line or draw an arrow above it, according to the inequality sign. For example, to graph x > 4, draw an open circle at 4 and shade the line to the right, or draw an arrow to the right. This represents all values greater than 4.

You can look at number lines in more detail here.

Drawing Graphical Inequalities

We can represent inequalities on a coordinate plane. For example, x > 4:

The dashed line represents x = 4, labelled as b and the shaded region, labelled as G, is x > 4.

To graph 2-variable linear inequalities on a coordinate plane:

Step 1: Rewrite the inequality in the slope-intercept form (y = mx + b), if necessary.

Step 2: Plot the line corresponding to the equality y = mx + b. Use a solid line for inclusive inequalities (≥, ≤) and a dashed line for strict inequalities (>, <). The solid line indicates that the points on the line are included in the solution set, while the dashed line shows that they are not.

Step 3: Choose a test point, usually the origin (0, 0), unless the line passes through it. Substitute the test point’s coordinates into the inequality. If the inequality holds true, shade the region containing the test point. If the inequality is not true, shade the region on the opposite side of the line.

For example, to graph y < 2x - 1:

1. The inequality is already in slope-intercept form.

2. Draw a dashed line representing y = 2x - 1, as the inequality is strict (y < 2x - 1).

3. Use the origin (0, 0) as the test point. Substitute its coordinates into the inequality: 0 < 2(0) - 1, which simplifies to 0 < -1. Since this statement is false, shade the region on the opposite side of the line, which is below the line in this case.

The graph of the inequality y < 2x - 1 consists of the dashed line y = 2x - 1 and the region below it.

Determining Graphical Inequalities

To find the equation of a line that has been drawn on a graph:

Step 1. Identify two points on the line: Locate any two distinct points on the line. If possible, choose points with integer coordinates for easier calculations.

Step 2. Calculate the slope: Find the slope (m) of the line by calculating the difference in y-coordinates divided by the difference in x-coordinates of the two points. The slope formula is: m = \frac{(y2 - y1)}{(x2 - x1)} where (x1, y1) and (x2, y2) are the coordinates of the two points.

Step 3. Find the y-intercept: The y-intercept (b) is the point where the line crosses the y-axis. If one of the points used in step 1 is on the y-axis, use its y-coordinate as the y-intercept. If not, you can substitute the slope (m) and one of the points (x1, y1) into the slope-intercept equation and solve for b: y1 = mx1 + b

Step 4. Write the equation of the line: Using the slope (m) and y-intercept (b), write the equation of the line in the slope-intercept form (y = mx + b).

Here’s an example:

Suppose you are given a graph with a line passing through the points (1, 3) and (3, 7).

1. Identify two points on the line: (1, 3) and (3, 7)

2. Calculate the slope: m = \frac{(7 - 3)}{(3 - 1)} = \frac{4}{2} = 2

3. Find the y-intercept:

  • Substitute the slope (m = 2) and one of the points (1, 3) into the equation: 3 = 2(1) + b
  • Solve for b: b = 3 - 2 = 1

4. Write the equation of the line: y = 2x + 1

So, the equation of the line is y = 2x + 1.

Examples

Example 1:

Graph the inequality y \geq -3x + 2.

1. Begin by graphing the line y = -3x + 2. To do this, first locate the y-intercept, which occurs when x = 0. In this case, the y-intercept is at point (0, 2).

2. Next, find another point on the line using the slope, which is -3. The slope represents the rise over run, or the change in y divided by the change in x. Since the slope is -3, this means the line goes down 3 units and moves to the right by 1 unit. From the y-intercept (0, 2), go down 3 units and to the right 1 unit to find the point (1, -1).

3. Now that you have two points, (0, 2) and (1, -1), draw a solid line connecting them. The solid line indicates that the points on the line are included in the solution set, as the inequality is inclusive (≥).

4. To determine which region to shade, choose a test point that does not lie on the line, such as the origin (0, 0). Substitute the coordinates of the test point into the inequality:

0 \geq -3(0) + 2

0 \geq 2

Since this statement is false, shade the region on the opposite side of the line, which is above the line in this case.

5. The graph of the inequality y \geq -3x + 2 consists of the solid line y = -3x + 2 and the region above it. Label the line, the y-intercept, and the two points used to draw the line for clarity. This graphical representation shows all the points (x, y) that satisfy the inequality y \geq -3x + 2.

Example 2:

Shade the region of a graph that satisfies the following inequalities simultaneously: y < x + 2, x < 1, and y > 1. After this, label the shaded region S.

1. Start by sketching or plotting the three inequalities on a coordinate plane:

a. y < x + 2: This is a straight line with a slope of 1 and a y-intercept of 2. Draw a dashed line representing this inequality, as the inequality is strict (y < x + 2).

b. x < 1: This is a vertical line passing through x = 1. Draw a dashed line representing this inequality as well, as the inequality is strict (x < 1).

c. y ≥ 1: This is a horizontal line passing through y = 1. Draw a solid line representing this inequality, as the inequality is inclusive (y \geq 1).

2. Determine the region that satisfies all three inequalities:

a. For y < x + 2, the region below the line y = x + 2 is shaded.

b. For x < 1, the region to the left of the line x = 1 is shaded.

c. For y > 1, the region above the line y = 1 is shaded.

3. Identify the region where all three shaded regions overlap. This is the region that satisfies all three inequalities simultaneously. In this case, it is a triangular region bounded by the lines y = x + 2, x = 1, and y = 1.

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