Problem 5 - Introductory Algebra for College Students

Question

Graph the solution set of each system of linear inequalities. $$\left\\{\begin{array}{l}x+3 y \leq 6 \\\x-2 y \leq 4\end{array}\right.$$

Step-by-Step Solution

Answer

The solution set is presented by the overlapping region of the graphs of the two linear inequalities. It encapsulates all points that satisfy both inequalities.

Step 1: Graph the First Inequality

Start by isolating y in the equation \(x + 3y \leq 6\). To do this, rewrite the inequality as \(3y \leq 6 - x\), and then divide every term by 3 to obtain \(y \leq 2 - \frac{1}{3}x\). This inequality represents a line with a slope of \(-\frac{1}{3}\) and a y-intercept of 2. Draw this line on the graph, making sure to use a solid line because the inequality symbol includes 'equal to' (≤). Shade the area below the line, since y is less than or equal to the expression \(2 - \frac{1}{3}x\).

Step 2: Graph the Second Inequality

Next, we need to graph the inequality \(x - 2y \leq 4\). Rewrite this as \(2y \geq x - 4\), and then divide by 2 to obtain \(y \geq \frac{1}{2}x - 2\). This line has a slope of \(\frac{1}{2}\) and a y-intercept of -2. Draw this line on the same graph, and shade the area above the line because y is greater than or equal to the expression \(\frac{1}{2}x - 2\).

Step 3: Solve the System of Inequalities

Finally, evaluate the system of inequalities by finding the overlapping region of the two shaded areas from Step 1 and Step 2. This region is the graph of the solution set that satisfies both inequalities.

Key Concepts

Graphing Inequalities

When dealing with systems of linear inequalities, one of the essential steps is to graph the inequalities on a coordinate plane. To do this, you first need to determine if the inequality line should be solid or dashed, depending on whether the inequality includes equality (≤ or ≥) or not (< or >).
Here, the equation for the first inequality was rewritten as \( y \leq 2 - \frac{1}{3}x \), showing a solid line because it includes "equal to".
Draw this line on the graph by finding the y-intercept (the point where the line crosses the y-axis) and plotting more points using the slope. Using the slope of -1/3, for each step right, move down one-third to find further points. Then connect the dots to form a straight line.
Once the line is drawn, shade below it because the condition is \( y \leq ... \), which means that the region representing the solution is below the line. This shaded area includes all the points \( (x, y) \) satisfying the inequality.
Repeat this process for each inequality to find where their solution regions overlap.

Solution Set

The solution set for a system of inequalities is the region where the solutions of the individual inequalities overlap. This is a key concept because only in this overlapping region do the solutions satisfy all the inequalities in the system simultaneously.
To find the solution set:

• Graph each inequality on the same coordinate plane, as described previously.
• Shade the correct side of each line as determined by the inequality sign (below/above the line).
• Identify the region that is shaded by both inequalities, as this represents the solution set.

For the exercise, the overlapping area of the shaded regions from the two inequalities \( y \leq 2 - \frac{1}{3}x \) and \( y \geq \frac{1}{2}x - 2 \) is the solution set. This area contains all the pairs \((x, y)\) that satisfy both inequalities.

Inequality Symbols

Inequality symbols are crucial in defining which region of the graph satisfies a linear inequality. The main symbols are:

• \(<\) and \(>\) - "less than" and "greater than"
• \(\leq\) and \(\geq\) - "less than or equal to" and "greater than or equal to"

In the context of graphing, these symbols determine whether the line representing the inequality should be drawn as a solid line (for \(\leq\) and \(\geq\)) or a dashed line (for \(<\) and \(>\)).
The symbol also helps to decide which side of the line to shade. For instance, \( y \leq 2 - \frac{1}{3}x \) means the area below the line is shaded, whereas with \( y \geq \frac{1}{2}x - 2 \), you shade above the line. Mastering the use of inequality symbols is vital as it allows you to accurately represent the range of solutions for each inequality.