Problem 25 - Elementary Algebra Within Reach

Question

In Exercises 23-28, sketch the graph of the system of linear inequalities. $$ \left\\{\begin{array}{lr} y \geq & -6 \\ y \leq & -8 x+9 \\ x \geq & 0 \\ y \leq & 0 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution is the area which includes parts of the fourth and second quadrants, bound by the line \(y = -6\), the line \(x = 0\), the line \(y = -8x + 9\) and the x-axis.

Step 1: Graphing Inequalities Individually

Begin with the first inequality \(y \geq -6\). Start by sketching the line for \(y = -6\). Because it's a 'greater than or equal to' inequality, shade upwards from the line. Repeat this process for the other inequalities: sketch the line for each, then shade the appropriate side of the line (upwards for \(y \geq -8x + 9\) and downwards for \(y \leq 0\)). For \(x \geq 0\), sketch the vertical line where x = 0 and shade to the right.

Step 2: Finding Overlap of the Inequality Regions

Now look for the region that's shaded on every graph. This area represents where all inequalities overlap, and is the solution to the system of inequalities.

Step 3: Representing the Solution

Finally, sketch the common shaded area on one graph, showing the encompassed region. This shaded area provides a visual representation of the solution to the system of inequalities.

Key Concepts

System of Inequalities

When dealing with a **system of inequalities**, we're looking at more than one inequality at the same time. Each inequality describes a specific set of solutions using linear equations, and when combined, the system narrows down to common solutions shared between all. Here's a quick guide for understanding systems of inequalities:

• Each inequality in the system will contribute its own condition that the solution must satisfy.
• The solution to such a system is not just one point but rather a region of the graph where all individual inequalities' conditions meet.

For example, in the given system:
- The inequality \(y \geq -6\) describes all points above or on the horizontal line \(y = -6\).
- The inequality \(y \leq -8x + 9\) includes all points on or below the line described by \(y = -8x + 9\).
- The inequality \(x \geq 0\) denotes all points on or to the right of the vertical line at \(x = 0\).
- Similarly, \(y \leq 0\) covers all points below or on the horizontal line \(y = 0\).

Together, these inequalities define a specific area on the graph where all these conditions overlap, representing the solution to the system.

Boundary Lines

**Boundary lines** play a vital role in graphing inequalities. They serve as the dividing lines on our graph and determine which areas satisfy the given inequalities. Here's how they function:

• A boundary line is derived from the equality part of an inequality. For example, for the inequality \(y \geq -6\), the boundary line is simply \(y = -6\).
• When an inequality is strict (like \(<\) or \(>\)), the boundary line is usually dashed, indicating that points on the line itself are not included.
• If the inequality includes equal to (\(\leq\) or \(\geq\)), the boundary line is solid, showing that the points on the line are part of the solution.

In our example, since inequalities such as \(y \geq -6\) and \(x \geq 0\) include the equals part (\(\geq\)), the lines will be solid. On a graph, these lines are essential, as they outline the boundaries within which solutions can lie once shaded appropriately.

Shading Regions

Shading is an essential step when graphing inequalities and helps to clearly visualize solutions. Once the boundary lines are drawn, the next step is shading the appropriate region:

• For each inequality, shade the area that satisfies it. For instance, if the inequality is \(y \geq -6\), you shade above the line \(y = -6\).
• The direction of shading can be determined by testing a point, usually the origin \((0,0)\), unless it's on the boundary, to see if it satisfies the inequality.
• The final solution region will be where all shaded areas overlap, indicating all conditions are met there.

In our exercise, different regions are shaded according to each inequality's requirement. For example, the region to the left of the line \(x = 0\) will not be shaded, as only \(x \geq 0\) is part of the solution. Only the overlapping shaded area shows the feasible solutions for the given system of inequalities, providing a clear visual representation of the solution set.