Problem 24 - Algebra and Trigonometry

Question

19–40 Graph the solution of the system of inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $$\left\\{\begin{aligned} x &>2 \\ y &<12 \\ 2 x-4 y &>8 \end{aligned}\right.$$

Step-by-Step Solution

Answer

Vertices are (2, -1); solution is unbounded below.

Step 1: Graph the basic inequalities

Start by graphing the inequality \(x > 2\). This is a vertical line at \(x = 2\), but since the inequality is strict, graph a dashed line showing all points to the right of the line.Next, graph the inequality \(y < 12\). This is a horizontal line at \(y = 12\), but since the inequality is strict, graph a dashed line showing all points below this line.

Step 2: Graph the linear inequality

Now, graph the line from the inequality \(2x - 4y > 8\). First, rewrite it in slope-intercept form: \[ 2x - 4y = 8 \rightarrow -4y = -2x + 8 \rightarrow y = \frac{1}{2}x - 2 \]This is a line with a slope of \(\frac{1}{2}\) and a \(y\)-intercept at \(-2\). Graph this line using a dashed line and shade the region above the line since the inequality is greater than (>).

Step 3: Determine the bounded region

Look for the intersection of the three shaded regions from Step 1 and Step 2. The solution set is where all inequalities overlap. It is the region bounded on the right by \(x = 2\), below by \(y = 12\), and above by the line \(y = \frac{1}{2}x - 2\).

Step 4: Find the vertices of the bounded region

Calculate the intersection points (vertices) of the lines:- Intersection of \(x = 2\) and \(y = \frac{1}{2}x - 2\): \( y = \frac{1}{2}(2) - 2 = -1. \) So, the vertex is \((2, -1)\).- Since \(y = 12\) does not intersect with \(y = \frac{1}{2}x - 2\) below 12, no intersection occurs.- Vertical asymptotes on the \(x = 2\) without intersection from \(2x - 4y > 8 \), confirm x-vertices if closed.Thus the lowest figure is open and unbounded at higher ranges.

Key Concepts

Linear Inequality

Understanding linear inequalities is essential for solving systems of inequalities. A linear inequality looks similar to a linear equation, but instead of an equal sign, it uses inequality signs such as <, >, ≤, or ≥. This means that a linear inequality doesn’t just show specific points but rather a range of solutions that falls within certain constraints. In the given exercise, we have the inequalities:

• \(x > 2\)
• \(y < 12\)
• \(2x - 4y > 8\)

These inequalities define constraints on a coordinate plane.
For instance, \(x > 2\) means we're looking for all the values where \(x\) is greater than 2, represented graphically with a dashed line moving to the right. The use of a dashed line signifies that the line itself is not included in the solution set, a detail that is crucial in graphing inequalities.

Bounded Region

The bounded region is a critical part of understanding the solution to a system of inequalities. A region is said to be bounded if it is enclosed within specific limits on all sides, defined by the inequalities. This means the solutions don’t extend infinitely in any direction. For the system in our problem, we identify the bounded region during the graphing of the inequalities. This is determined by the overlapping or intersection of the shaded regions, where:

• \(x > 2\) is bounded on the left.
• \(y < 12\) provides the upper boundary.
• \(y = \frac{1}{2}x - 2\) encloses the area above.

When these conditions are met simultaneously, the solution set is fully enclosed by these boundaries, although in this problem, the extent becomes open and unbounded in terms of certain possible ranges, particularly when extending beyond the limits graphically.

Graphing Inequalities

Graphing a system of inequalities involves plotting each inequality on a coordinate graph to determine where their solution sets intersect. This graphical area represents the solution to the system of inequalities. To graph an inequality:

• Start with one inequality at a time.
• Use dashed or solid lines – dashed if it does not include the boundary line (\( > \) or \( < \)), and solid if it does (\( \geq \) or \( \leq \)).
• For our exercise, the lines are dashed since all inequalities are strict (do not include the boundary).
• Shade the correct side of the line that includes the possible solutions.

The intersection or overlap of all these shaded regions gives the solution set. The vertices, or corner points, where the boundaries intersect help identify specific points in the solution. In our example, the graphing starts with the inequalities \(x > 2\) and \(y < 12\), and continues with graphing the line \(2x - 4y > 8\), rewritten as \(y = \frac{1}{2}x - 2\). The final solved graph is where all these shaded areas cross.