Problem 9 - A Graphical Approach to College Algebra

Question

Graph each inequality. Do not use a calculator. $$5 x \leq 4 y-2$$

Step-by-Step Solution

Answer

Shade the area above the line \( y = \frac{5}{4}x + \frac{1}{2} \), including the line itself.

Step 1: Rewrite the Inequality

First, we need to rearrange the inequality so that it resembles the slope-intercept form (\( y = mx + b \)). Begin by solving the inequality for \( y \). Start with the given inequality: \[ 5x \leq 4y - 2 \]Add 2 to both sides of the inequality:\[ 5x + 2 \leq 4y \]

Step 2: Isolate \( y \)

Next, divide every term by 4 to solve for \( y \):\[ y \ge \frac{5}{4}x + \frac{1}{2} \] The inequality is now in the "greater than or equal to" slope-intercept form.

Step 3: Identify the Slope and Y-Intercept

Now that the equation is in slope-intercept form \( y = mx + b \), identify the slope (\( m \)) and the y-intercept (\( b \)). Here, the slope (\( m \)) is \( \frac{5}{4} \), and the y-intercept (\( b \)) is \( \frac{1}{2} \).

Step 4: Draw the Boundary Line

Use the slope and y-intercept to plot the boundary line on the graph. Start by plotting the y-intercept \((0, \frac{1}{2})\) on the y-axis. From there, use the slope to find another point. With a slope of \( \frac{5}{4} \), you move up 5 units and right 4 units from the y-intercept to get the second point. Draw a solid line through these points because the inequality includes \( y \) being equal to \( (\geq) \).

Step 5: Shade the Solution Region

Since the inequality is \( y \geq \frac{5}{4}x + \frac{1}{2} \), shade the region above and including the line. This shows all the values \( (x, y) \) that satisfy the inequality, where \( y \) is greater than or equal to the line.

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form is fundamental when it comes to graphing linear equations and inequalities. This form allows you to instantly read the slope and y-intercept of a line, which are crucial for plotting. The slope-intercept form of a line is given by the equation: \[ y = mx + b \] where:

• \( m \) is the slope of the line, which depicts the steepness and direction. It is calculated as the rise over run, showing how much \( y \) changes for each unit increase in \( x \).
• \( b \) is the y-intercept, the point where the line crosses the y-axis. It tells where the line starts when \( x = 0 \).

By rearranging an inequality or equation into this form, you can readily get values needed for graphing. This makes the slope-intercept form incredibly useful and popular for understanding linear relationships.

Linear Inequalities

Linear inequalities extend the concept of linear equations. Instead of expressing equality, they show a range of possible solutions, marked by inequality symbols such as \( >, <, \geq \), and \( \leq \). A linear inequality like \( y \ge \frac{5}{4}x + \frac{1}{2} \) tells us not only about the slope and y-intercept of a boundary line but also provides information on which side of this line forms the solution set.

There are key things to remember when working with linear inequalities:

• Inequalities can be thought of as similar to equations, except you shade a region instead of drawing just a line.
• The boundary line itself might be solid or dashed, depending on the inequality sign. "Greater than or equal to" (\( \geq \)) and "less than or equal to" (\( \leq \)) include the line as part of the solution, so a solid line is used.
• When dividing or multiplying through an inequality by a negative number, the inequality sign flips.

Understanding these inequalities helps visualize solutions that include all points satisfying the given conditions, adding a dynamic aspect to graphing.

Graphing Techniques

Graphing techniques are vital for translating algebraic inequalities into visual forms. Here, the main goal is to create clear graphs that show all possible solutions. Below are steps to graphing an inequality:

• First, convert the inequality into slope-intercept form, if it is not already. This enables easier plotting.
• Identify the slope and y-intercept from the equation.
• Plot the y-intercept on the graph. For instance,\((0, \frac{1}{2})\) is where you start in our example.
• Next, use the slope to find other points. Given \( \frac{5}{4} \) as the slope, move 5 units up and 4 units to the right from the y-intercept to locate another point.
• Draw the boundary line. Since the inequality in our example is "greater than or equal" (\( \geq \)), this line will be solid.
• Finally, shade the region that represents the solution set. This region will be above the line for the given example, as the inequality is \( y\) is greater than or equal to.

These techniques transform algebraic expressions into visual representations, providing a clear picture of possible solutions.