Problem 78 - Algebra 1

Question

GRAPHING Graph the system of linear inequalities. $$ \begin{aligned} &x+2 y<3\\\ &x-3 y>1 \end{aligned}$$

Step-by-Step Solution

Answer

The solution to the given system of inequalities is the overlapping area from shading below the line \(y=\frac{3-x}{2}\) and above the line \(y=\frac{x-1}{3}\).

Step 1: Convert inequalities

Convert the inequalities into 'y' form. The first one is \(x+2y<3\), subtract 'x' from both sides to get \(2y<3-x\), divide by 2 to get \(y<\frac{3-x}{2}\). Do the same with the second inequality to get \(y<\frac{x-1}{3}\).

Step 2: Graphing inequalities

Start graphing these inequalities. First, plot the line of the first inequality \(y=\frac{3-x}{2}\). The slope is -1/2 and the y-intercept is 1.5. Next, plot the line for the second inequality \(y=\frac{x-1}{3}\). The slope is 1/3 and the y-intercept is -1/3. According to the inequalities, \(y<\frac{3-x}{2}\) and \(y>\frac{x-1}{3}\). This means for the first line we will shade below and for the second line we will shade above.

Step 3: Identify solution area

The solution of the system of inequalities is where the shaded areas intersect. The intersection of the two shadings represents all the points (x, y) that satisfy both inequalities simultaneously.

Key Concepts

System of Inequalities

A system of inequalities involves more than one inequality being solved or graphed together. When dealing with a system of linear inequalities like \( x+2y<3 \) and \( x-3y>1 \), you're looking for all the solutions that satisfy both inequalities at the same time.
These inequalities are often graphed to find the solution region. The solution to the system is represented by the area where the shaded regions of each inequality overlap.
Graphically solving a system of inequalities involves plotting each inequality on the same coordinate grid and identifying the shared region.
Understanding a system of inequalities is crucial in visualizing problems involving constraints and conditions in real-world scenarios.

Slope-Intercept Form

The slope-intercept form is a way of writing the equation of a line using its slope and y-intercept. It is written as \( y = mx + b \), where \( m \) represents the slope and \( b \) represents the y-intercept.
For the inequality \( x+2y<3 \), after converting it to slope-intercept form, it becomes \( y < \frac{3-x}{2} \). Here, the slope \( m \) is -1/2, and the y-intercept \( b \) is 1.5.
Similarly, the second inequality \( x-3y>1 \), can be expressed as \( y < \frac{x-1}{3} \), with a slope of \( 1/3 \) and a y-intercept of -1/3.
Converting to this form makes it easier to graph the equations and understand their behavior, as you can quickly identify how steep the line is and where it crosses the y-axis.

Shading Regions

Shading regions on a graph is a technique used to visually represent inequality solutions. After plotting the boundary lines of the inequalities, shading helps indicate which side of the line contains the solutions.
For the inequality \( y < \frac{3-x}{2} \), the region below the line is shaded, signifying that all values of \( y \) beneath the line fulfill this inequality.
For the second inequality \( y > \frac{x-1}{3} \), shade above the line, indicating that values above satisfy this inequality.
The intersection where the shading from both inequalities overlaps is the solution area. This overlapping region encompasses all the points that solve the system of inequalities.

• Identify each inequality's boundary line.
• Determine which side of the line to shade.
• Find the overlapping shaded region to get the solution to the system.

This technique provides a clear visual representation, making it easier to understand which solutions work for both inequalities in the system.