Problem 59 - College Algebra Essentials

Question

Graph the solution set of each system of inequalities or indicate that the system has no solution. $$ \left\\{\begin{array}{l} x \geq 0 \\ y \geq 0 \\ 2 x+5 y<10 \\ 3 x+4 y \leq 12 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution set will be the intersection of all the regions shaded by the inequalities, which will be a specific area in the first quadrant since \(x \geq 0\) and \(y \geq 0\).

Step 1: Graph the Equalities

Start by ignoring the inequality signs and plotting the lines \(x = 0\), \(y = 0\), \(2x+5y = 10\) and \(3x+4y = 12\).

Step 2: Add Inequality Nature on Graph

For each inequality, determine which side of the line to shade. For \(x \geq 0\), shade to the right. For \(y \geq 0\), shade upwards. For \(2x+5y < 10\) and \(3x+4y \leq 12\), you'll need to pick a test point. If it satisfies the inequality, shade that side. If it doesn't, shade the other side.

Step 3: Identify Solution Region

The solution will be the intersection of all the shaded regions. This will tell you the range of x and y values that satisfy all the inequalities.

Key Concepts

Graphing Inequalities

When dealing with systems of inequalities, graphing is a powerful visual tool. First, we graph each equation as if they were equalities, essentially drawing boundary lines. For the given system, start with the lines:

• Vertical line at \(x = 0\)
• Horizontal line at \(y = 0\)
• Slanted lines following the equations \(2x + 5y = 10\) and \(3x + 4y = 12\)

These lines break down the coordinate plane into various regions. Notice their slopes and intercepts help situate them precisely. When using inequalities, these lines include or exclude regions based on their inequality symbols. So essentially, your graph will display areas that meet the conditions set by these inequalities.

Solution Set

The solution set represents all possible values of \(x\) and \(y\) that satisfy the system of inequalities. Imagine it as a hyper-multi-dimensional area on the graph.Once you graph each inequality properly, navigate these steps to locate the solution set:

• Observe which areas all shaded regions overlap
• Highlight where conditions like \(x \geq 0\) and \(y \geq 0\) intersect with others like \(2x+5y<10\) and \(3x+4y \leq 12\)

The final solution set is the area that survives each of these conditions. It's this shared portion that fulfills every inequality in the system.

Inequality Shading

Shading plays a critical role in identifying the solution space for a system of inequalities. Each inequality is associated with a particular half-plane. Follow these steps to shade effectively:

• If the inequality is "greater than" (\(x \geq 0\) or \(y \geq 0\)), shade on the side that extends beyond the axis.
• For inequalities like \(2x+5y < 10\) or \(3x+4y \leq 12\), use a test point not on the boundary line, like the origin (\(0,0\)), to determine the correct side to shade. If the test point meets the inequality, the side where it lies should be shaded.

The shaded part represents potential solutions. Always pay attention to whether your inequality is strict (\(<\), \(>\)) or inclusive (\(\leq\), \(\geq\)) as this will dictate whether border lines are solid or dashed, showing boundaries within or beyond which solutions lie.

Intersection of Inequalities

In the context of graphing systems of inequalities, finding the intersection means identifying the area where all the conditions are true at once. This intersection is your holy grail – the definitive solution area to these inequalities. Here's how to pinpoint the intersection:

• Look at the graph where all the shaded parts from each inequality overlap
• This area is permissible for every inequality in the set
• Verify that this intersection aligns with the logic of each inequality's constraints

This overlapping region, sometimes a polygonal shape, showcases the solution space. Each intersecting point or area within it adheres to every single condition laid by the inequality system, indicating harmony across all equations.