Problem 14 - Precalculus: Functions and Graphs

Question

Sketch the graph of the system of Inequalities. $$\left\\{\begin{array}{c}2 y-x \leq 4 \\\3 y+2 x<6\end{array}\right.$$

Step-by-Step Solution

Answer

Shade the region below the line \(2y = x + 4\) and below the line \(3y = -2x + 6\).

Step 1: Convert inequality to equation

We begin by converting the inequality into an equation. Starting with the first inequality, replace the ≤ sign with an equality: \(2y - x = 4\).

Step 2: Determine the line for the first equation

To determine this line, we need two points. Set \(x = 0\), compute \(y\): \(2y = 4\) leads to \(y = 2\). So, the point (0, 2) is on this line. Now set \(y = 0\), compute \(x\): \(-x = 4\) which gives \(x = -4\), thus the point (-4, 0) is on this line.

Step 3: Convert the second inequality to an equation

Change the second inequality to an equation by replacing < with =: \(3y + 2x = 6\).

Step 4: Determine the line for the second equation

Using similar steps as Step 2, to get two points on the line, firstly set \(x = 0\), compute \(y\): \(3y = 6\) results in \(y = 2\), so (0, 2) is a point on this line. Next, set \(y = 0\), compute \(x\): \(2x = 6\) leading to \(x = 3\), thus (3, 0) is another point on the line.

Step 5: Sketch the graph of both lines

Draw both lines on a graph. The line \(2y - x = 4\) will pass through (-4, 0) and (0, 2). The line \(3y + 2x = 6\) will pass through (0, 2) and (3, 0). Both lines meet at the point where they have the same coordinates.

Step 6: Determine the shading region for the solution

For the first inequality \(2y - x \leq 4\), test the origin (0,0): \(2(0) - (0) = 0 \leq 4\), so the area including (0,0) satisfies the inequality. For the second inequality \(3y + 2x < 6\), test the origin: \(3(0) + 2(0) = 0 < 6\), which also passes. Shade the region including the origin and in common to both half-planes.

Key Concepts

Linear Inequalities

Linear inequalities are similar to linear equations but instead of an equality sign, they use inequality signs such as <, >, ≤, or ≥. These inequalities define a set of solutions that form a region on a graph, rather than just a single line.
For example, in the inequality \(2y - x \leq 4\), any point \((x, y)\) that makes this inequality true is part of the solution set.

• This means the graph will include a shaded area representing all the possible solutions.
• The line \(2y - x = 4\) acts as a boundary for this region.
• Since it's \(\leq\), points on the line are also valid solutions, hence the line is solid.

To determine the boundary line of the inequality, replace the inequality sign with an equal sign, which allows us to graph the line in the coordinate plane.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This helps us graph equations, such as linear inequalities, in a two-dimensional space.

• Each point on the graph is represented by an ordered pair \((x, y)\) which corresponds to its location along the x-axis and y-axis.
• The coordinate plane divides space into four quadrants which help us describe the position of points.
• By plotting these points, we can draw lines, curves, and shaded regions to visually represent solutions to inequalities and systems of equations.

In our exercise:

• The line \(2y - x = 4\) is determined by the points \((-4, 0)\) and \((0, 2)\).
• The line \(3y + 2x = 6\) passes through the points \((0, 2)\) and \((3, 0)\).

By using these points, you can easily graph the lines and identify the regions they describe.

Systems of Inequalities

When we have more than one inequality, we are dealing with a system of inequalities. Solving these means finding the set of points that satisfy all the inequalities simultaneously.

• Each inequality in the system divides the plane into two regions: one where it is satisfied and one where it is not.
• The solution to the system is the intersection of these regions, typically resulting in a polygonal-shaped solution area.
• To find the solution area, you test points or simply observe where the shaded areas from all inequalities overlap.

In the system given:

• The overlap of the regions derived from the inequalities \(2y - x \leq 4\) and \(3y + 2x < 6\) needs to be shaded.
• Checking a test point, like the origin \((0,0)\), helps confirm which regions to shade.

Thus, graphing systems of inequalities allows us to visualize complex solution spaces, providing a clear understanding of where solutions exist for all included inequalities.