Problem 314 - Elementary Algebra

Question

$$ \left\\{\begin{array}{l} y<\frac{3}{4} x-2 \\ -3 x+4 y<7 \end{array}\right. $$

Step-by-Step Solution

Answer

Graph the inequalities and find the overlapping shaded region where both conditions are satisfied.

Step 1: Understand the Inequalities

We have two inequalities here: 1. \( y < \frac{3}{4} x - 2 \) 2. \( -3x + 4y < 7 \). Each inequality represents a region in the coordinate system.

Step 2: Graph the First Inequality

Start by graphing the boundary line for the first inequality, which is \( y = \frac{3}{4} x - 2 \). This is a straight line. Then, shade the region below the line because the inequality is less than (\( < \)).

Step 3: Graph the Second Inequality

Next, convert \( -3x + 4y < 7 \) to its boundary line by setting it as an equation: \( -3x + 4y = 7 \). Graph this line and then shade the region below it since the inequality is also less than (\( < \)).

Step 4: Identify the Solution Region

The solution to the system is the region where the shaded areas from both inequalities overlap. This is the area that satisfies both inequalities simultaneously.

Key Concepts

Linear Inequalities

A linear inequality is similar to a linear equation, but instead of an equal sign, it uses inequality symbols such as <, >, ≤, and ≥. These inequalities describe regions on a coordinate plane, not just lines.
For example, the inequality \( y < \frac{3}{4} x - 2 \) represents all points below the line \( y = \frac{3}{4} x - 2 \). When graphing linear inequalities, always start by graphing the corresponding boundary line (the equation). If the inequality is < or >, draw a dashed line, which shows that points on the line are not included. For ≤ or ≥, a solid line is drawn, indicating the points on the line are included in the solution set. Afterward, shade the region above or below the line based on the inequality symbol in the inequality.

System of Inequalities

A system of inequalities consists of more than one inequality that must be satisfied simultaneously by the same set of variables. In this exercise, we are given two inequalities:
\( y < \frac{3}{4} x - 2 \) and \( -3x + 4y < 7 \). Each inequality is graphed separately to find where their solution sets overlap.
The process of solving a system involves graphing each inequality on the same set of axes and then identifying the region of overlapping shaded areas. This overlapping region is the solution set for the system.
Here's a quick summary for graphing a system of inequalities:
- Graph each inequality one at a time.
- Use dashed or solid lines based on the inequality symbols.
- Shade the appropriate region (above or below the line) for each inequality.
- The solution to the system is the intersection (overlapping area) of the shaded regions.

Graphical Representation

Graphical representation is a visual method of solving mathematical problems, especially useful for inequalities and systems of inequalities. The approach simplifies understanding by providing a visual context. In our exercise, we graph lines to represent the boundaries of each inequality and shade the corresponding regions.
To graph the boundary line for \( y < \frac{3}{4} x - 2 \):
- Find the intercepts and plot them.
- Draw a dashed line as the boundary.
- Shade the region below the line since y is less than the line's equation.
Similarly, for the boundary line \( -3x + 4y < 7 \):
- Convert to standard form and find intercepts.
- Graph a dashed line for this inequality.
- Shade the region below the line.
The final solution is the intersection of these shaded regions, which can help us visually determine where both inequalities are satisfied.