Problem 6 - A Graphical Approach to College Algebra

Question

Graph each inequality. Do not use a calculator. $$4 y-3 x<5$$

Step-by-Step Solution

Answer

Graph the dashed line \( y = \frac{3}{4}x + \frac{5}{4} \) and shade below it.

Step 1: Rewrite the Inequality

First, we should rewrite the inequality in a form that helps us understand the boundary line. To do this, isolate the boundary equation as if it were an equality: \( 4y - 3x = 5 \).

Step 2: Express y in Terms of x

Convert the boundary equation into the slope-intercept form, \( y = mx + b \). Start by moving the \(-3x\) term to the other side: \( 4y = 3x + 5 \). Then divide everything by 4 to solve for \( y \): \[ y = \frac{3}{4}x + \frac{5}{4} \].

Step 3: Identify the Boundary Line

The equation \( y = \frac{3}{4}x + \frac{5}{4} \) represents the boundary line, which will be a straight line with a slope of \( \frac{3}{4} \) and a y-intercept of \( \frac{5}{4} \).

Step 4: Determine Line Type and Graph

Since the inequality is \( 4y - 3x < 5 \) (not \( \leq \)), the boundary line on the graph will be dashed to indicate that points on the line are not solutions to the inequality. Graph this dashed line accordingly.

Step 5: Determine the Shaded Area

To find out which side of the boundary line to shade, we need to test a point. A common choice is the origin \( (0, 0) \) if it is not on the line. Substitute \( x = 0 \) and \( y = 0 \) into the original inequality: \( 4(0) - 3(0) < 5 \), which simplifies to \( 0 < 5 \). This is true, so we shade the area below and left of the line that includes the origin.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is a powerful tool for graphing. It is represented as \( y = mx + b \). Here, \( m \) is the slope of the line, which describes how steep the line is. The \( b \) is the y-intercept, the point where the line crosses the y-axis.
To convert an equation to this form, you need to solve for \( y \) in terms of \( x \). For example, from our boundary line \( 4y - 3x = 5 \), we moved the \( -3x \) to the other side and divided by 4. This gave us: \( y = \frac{3}{4}x + \frac{5}{4} \).

• **Slope (m):** Tells how the line 'rises' or 'falls' as you move along the x-axis.
• **Y-Intercept (b):** Tells where the line meets the y-axis.

These will help you quickly sketch the line by starting at the y-intercept \((0, b)\) and using the slope \( m \) to find other points on the line.

Boundary Line

In the context of graphing inequalities, the boundary line helps distinguish which area of the graph contains solutions. The boundary line itself comes from rewriting the inequality as an equality. It's a visual divider. For instance, in the inequality \( 4y - 3x < 5 \), the boundary is drawn from the equation \( y = \frac{3}{4}x + \frac{5}{4} \).
When graphing:

• The **line doesn't include solutions** as it comes from an inequality (e.g., \(<\), not \(\leq\)).
• It's used to determine which region should be shaded.

The boundary line delineates regions where the inequality holds true from those where it doesn’t. When testing points in these regions, solutions fall in the shaded part, while boundary points are not included in \(<\) or \(>\) inequalities.

Dashed Line

A dashed line can be seen on the graph when the inequality is strict, such as \(<\) or \(>\). This indicates that points on the line are not part of the solution set. For our inequality \( 4y - 3x < 5 \), when we convert this to a line \( y = \frac{3}{4}x + \frac{5}{4} \), the relevant line on the graph will be dashed.
Here’s why it's important:

• **Visual Clue:** It reminds you not to include points lying exactly on the line as solutions.
• **Contrast With Solid Lines:** Solid lines are used when \(\leq\) or \(\geq\) include the boundary in solutions.

By drawing a dashed line, it’s clear that the boundary isn't included, making your solution to the inequality easy to interpret.

Shaded Region

The shaded region of a graph demonstrates where the solutions to an inequality lie. Once the boundary line is drawn (as a dashed line or solid line), the next step is to find which side of the line to shade. This shows all the points \((x,y)\) that satisfy the inequality.
For \( 4y - 3x < 5 \), after finding and drawing the boundary, you choose a test point to check on which side of this line the inequality holds true.

• **Common Test Point:** Often \((0,0)\) is used unless it lies on the line.
• **Check Expression:** Substitute into the original inequality. If true, shade that side; if false, shade the opposite.
• **Clarity of Solutions:** The shaded region clearly separates the feasible solutions in the graph.

For this inequality, after testing point \((0,0)\), the region including the origin is shaded, showing all points less than the boundary line.