Problem 36 - Intermediate Algebra

Question

Graph each inequality. $$ \frac{x}{3}-\frac{y}{2} \geq 1 $$

Step-by-Step Solution

Answer

Graph the line \( y = \frac{2}{3}x - 2 \) and shade the region above it.

Step 1: Write the Inequality as an Equation

To graph the inequality \( \frac{x}{3} - \frac{y}{2} \geq 1 \), first treat the inequality as an equation for simplicity: \( \frac{x}{3} - \frac{y}{2} = 1 \). This will help us find the boundary line.

Step 2: Convert to Slope-Intercept Form

We want to express the equation \( \frac{x}{3} - \frac{y}{2} = 1 \) in the form \( y = mx + b \). Rearrange as follows:1. Multiply every term by 6 to eliminate fractions: \[ 2x - 3y = 6 \]2. Isolate \( y \): \[ -3y = -2x + 6 \] Divide by -3: \[ y = \frac{2}{3}x - 2 \]

Step 3: Graph the Line

Graph the equation \( y = \frac{2}{3}x - 2 \):1. Start at the y-intercept \((0, -2)\).2. Use the slope \( \frac{2}{3} \) to find another point: from \((0, -2)\), move up 2 (rise) and right 3 (run) to get to \((3, 0)\).3. Draw a solid line through these points because the original inequality is \( \geq \), which includes the line.

Step 4: Shade the Solution Region

Since the inequality is \( \geq \) (greater than or equal to), shade the region above the line. You can test a point that is not on the line, like \( (0,0) \), to ensure it's not in the solution region:- Substitute \( (0,0) \) into \( \frac{x}{3} - \frac{y}{2} \geq 1 \): \( \frac{0}{3} - \frac{0}{2} = 0 \), which is not \( \geq 1 \).The region where \( (0,0) \) is located should not be shaded. Thus, the area to be shaded is the one above the line.

Key Concepts

Slope-Intercept Form

For those grappling with graphing inequalities, the slope-intercept form is your trusty friend. It's the format \( y = mx + b \) where:

• \( m \) is the slope of the line, indicating how steep it is.
• \( b \) is the y-intercept, showing where the line crosses the y-axis.

When you have an inequality like \( \frac{x}{3} - \frac{y}{2} \geq 1 \), it's often easier to start by turning it into an equation, \( \frac{x}{3} - \frac{y}{2} = 1 \). After that, rearranging to slope-intercept form makes graphing a breeze. Here's a simple trick: multiply every term by 6 to nix those fractions, giving you \( 2x - 3y = 6 \). Now, isolate \( y \) to get \( y = \frac{2}{3}x - 2 \). This tells us that the slope, \( \frac{2}{3} \), rises 2 units for every 3 units it runs to the right, and the line crosses the y-axis at \(-2\).
Next time you see a linear inequality, think slope-intercept form. It's like reading a road map of the line.

Boundary Line

The boundary line is what you get when you set an inequality as an equation. It divides the graph into two parts. For \( \frac{x}{3} - \frac{y}{2} = 1 \), this boundary line is what keeps one side of the graph separated from the other.
Imagine the graph as a vast land, and the boundary line is like a fence. In our case, this fence is the line \( y = \frac{2}{3}x - 2 \). Drawing this boundary line provides a visual aid to understand the inequality better.

• If the inequality is \( \geq \) or \( \leq \), the boundary line is solid. This means that points on the line satisfy the inequality.
• Conversely, if the inequality was \( > \) or \( < \), it would be dashed because the line itself would not be included.

While the inequality \( \frac{x}{3} - \frac{y}{2} \geq 1 \) includes the line, the graph of \( y = \frac{2}{3}x - 2 \) in this instance is solid. It represents the locations where the inequality holds exactly at equal, sort of like a tightrope exactly on the edge.

Solution Region

The solution region is the heart of graphing inequalities; it's the set of all points that satisfy the inequality. Once the boundary line is up, it's time to decide which side holds the solutions.
When graphing \( \frac{x}{3} - \frac{y}{2} \geq 1 \), the inequality's direction \( \geq \) indicates we are interested in the area above the line. Think of it as shading the region where flying kites would be allowed, above the tightrope. How do we know this shading is correct?

• Pick a test point not on the boundary line, like \((0, 0)\). This point will help check which side of the line satisfies the inequality.
• Place \((0, 0)\) into the inequality: \( \frac{0}{3} - \frac{0}{2} = 0 \) which is not \( \geq 1 \).

This result means \((0, 0)\) is not in the solution region, so the part of the plane where the value \(< 1 \) isn't shaded. Instead, we shade the opposite section.
Understanding the solution region means recognizing which half of the graph explains the inequality. This logical process will illuminate your path through inequalities.