Problem 24 - Algebra 1

Question

Graph the system of linear inequalities. \(2 x+y \geq 2\) \(x \leq 3\) \(y \leq \frac{1}{2}\)

Step-by-Step Solution

Answer

The solution to the system of linear inequalities is the intersecting region that satisfies all three inequalities: \(2x + y \geq 2\), \(x \leq 3\), and \(y \leq 0.5\).

Step 1: Graph each inequality individually

We start by plotting each inequality individually. The inequality \(2x + y \geq 2\) can be rewritten in slope-intercept form \(y = -2x + 2\). When graphed, this will be a line that intersects the y-axis at \(y = 2\) and has a downward slope of -2. As the inequality is 'greater than or equal to', the region above the line will be shaded. The inequality \(x \leq 3\) is a vertical line crossing the x-axis at \(x = 3\). As this is 'less than or equal to', the area to the left of the line will be shaded. Similarly, the inequality \(y \leq 0.5\) is a horizontal line crossing the y-axis at \(y = 0.5\). Here, the area below the line will be shaded because the inequality is 'less than or equal to'.

Step 2: Identify the intersection

To identify the solution to the system of inequalities, locate the region where all shaded areas intersect. This region is the one that fulfills all three inequalities simultaneously.

Step 3: Draw the final graph

Finally, draw the final graph highlighting the intersecting region. This is the solution to the problem.

Key Concepts

Graphing Inequalities

Graphing inequalities is like plotting a boundary represented by a line on a graph. This line divides the plane into two halves. For any inequality, the area on one side of the line will satisfy the inequality, while the other will not. When graphing, always consider:

• Converting the inequality into an equation by replacing the inequality sign with an equals sign.
• This allows you to plot the boundary line. It's important to note whether the line is solid or dashed. A solid line means the inequality includes 'equal to' (≥ or ≤), indicating the points on the line make the inequality true.
• Conversely, a dashed line means the inequality is strict (>, <), excluding points exactly on the line.
• Once the line is complete, determine which side of it satisfies the inequality by picking a test point (often the origin, unless the line passes through it).
• Shade the area that meets the inequality condition.

Intersection of Inequalities

The intersection of inequalities is the area that fulfills all given conditions in a system of inequalities. Think of it as finding the common area that overlaps between all shaded regions from individual inequalities. Here's a simple way to approach it:

• After shading each inequality on the graph, look for the places where the colored areas overlap.
• This overlapping area represents the set of points that satisfy the entire system of inequalities.
• In the context of the exercise "Graph the system of linear inequalities", this means finding the common area where the plotted regions of all three inequalities meet.
• Only points in this intersection will be solutions to the system.

This intersection ensures that all conditions from the inequalities are true at once, providing a visual representation of solutions.

Linear Inequality Solution

A linear inequality solution represents all possible values that satisfy an inequality. When working with systems of inequalities, like the ones in the original exercise, solutions exist only in the region where all inequalities intersect. Here’s how you get to this solution:

• First, make sure each inequality is converted and graphed correctly, paying attention to the correct boundary line type (solid or dashed).
• Shade appropriately based on the inequality sign (greater than or equal to, less than or equal to, etc.).
• Identify the intersection area, which comes from combining all shaded regions on the same graph.
• Every point within this intersection area satisfies all included inequalities, making it part of the solution set.

Understanding this helps in determining feasible regions in optimization problems or when analyzing constraints in real-world scenarios.