Problem 31 - Intermediate Algebra

Question

Graph each inequality. $$ y \geq \frac{8}{3} $$

Step-by-Step Solution

Answer

Graph a solid line at \( y = \frac{8}{3} \) and shade above it.

Step 1: Understand the Inequality

The inequality given is \( y \geq \frac{8}{3} \). This means that \( y \) can take any value that is equal to or greater than \( \frac{8}{3} \).

Step 2: Plot the Line \( y = \frac{8}{3} \)

On a coordinate plane, draw a horizontal line at \( y = \frac{8}{3} \). This line represents all the points where \( y \) equals \( \frac{8}{3} \).

Step 3: Determine the Shading Area

Since the inequality is \( y \geq \frac{8}{3} \), shade the region above the line. This includes the line itself and extends upwards, indicating all values of \( y \) that are greater than \( \frac{8}{3} \).

Step 4: Check the Boundary

The line \( y = \frac{8}{3} \) is part of the solution set because the inequality is 'greater than or equal to' (\( \geq \)). Thus, the line should be drawn solid, not dashed.

Key Concepts

Coordinate Plane

The coordinate plane is a fundamental concept in graphing, providing a two-dimensional surface where points are plotted using a pair of numerical coordinates. These coordinates determine the position of a point on the plane: one tells us the horizontal position (x-axis), and the other describes the vertical position (y-axis). Without a coordinate plane, graphing inequalities and equations wouldn't be systematic or visually intuitive.

• X-Axis: The horizontal line that extends infinitely in both directions.
• Y-Axis: The vertical line that also extends infinitely upwards and downwards.
• Origin: The point where the axis intersect, represented as (0, 0).

When dealing with inequalities like in our exercise, 'y' is compared with a constant value, resulting in a horizontal line across this plane. Here, it acts as a visual boundary for the area of interest.

Shading Regions

In graphing inequalities, shading is crucial as it indicates which part of the graph contains solutions to the inequality. The shaded region encompasses all possible solutions, and selecting the correct area to shade depends on the inequality's nature.

• Solid Line and Area Above: For inequalities such as \( y \geq c \), the shading is above the line, including the line itself.
• Dashed Line and Area Below: For \( y < c \) or similar, shade below the line, excluding the line.

In this exercise, since we have \( y \geq \frac{8}{3} \), we shade above the line. This represents all the points where 'y' is equal to or greater than \( \frac{8}{3} \). Proper shading communicates the complete set of possible solutions visually.

Inequality Solutions

Inequalities are statements about the relative size of two expressions. In a graph, they signify which side of a line contains solutions. Solutions to inequalities are more than mere numbers—they represent ranges of values satisfying the inequality.

For \( y \geq \frac{8}{3} \), any 'y' value on or above the line satisfies the inequality. This can be tested by selecting points and substituting them into the inequality:

• Includes Points: Any point like (0, 3) works, as 3 is more than \( \frac{8}{3} \).
• Boundary Test: The line itself is in the solution set, so (0, \(\frac{8}{3}\)) is valid.
• Excludes Points: But (0, 2) isn't a solution, as 2 is less than \( \frac{8}{3} \).

Knowing where to shade based on inequality helps students understand which 'y' values solve the inequality, enhancing comprehension of solution sets.