Problem 22 - Algebra 1: Concepts and Skills

Question

Graph the system of linear inequalities. $$ \begin{aligned} &x>-2\\\ &y \geq-2\\\ &y<4 \end{aligned} $$

Step-by-Step Solution

Answer

The solution to these inequalities is the region between the lines \(y=-2\) and \(y=4\), to the right of the line \(x=-2\).

Step 1 - Plotting the line \(x > -2\)

To graph \(x > -2\), a vertical line is drawn at \(x = -2\), but since the inequality is strictly greater than, there should be no line at \(x = -2\). Therefore, dashed line is drawn.

Step 2 - Plotting the line \(y \geq -2\)

Next, to graph \(y \geq -2\), a horizontal line is drawn at \(y = -2\). Since the inequality is greater than or equal to, the line at \(y = -2\) will be solid, not dashed.

Step 3 - Plotting the line \(y < 4\)

The last inequality to graph is \(y < 4\). This is another horizontal line, but this time at \(y = 4\). Here the inequality is strictly less than, so a dashed line is drawn.

Step 4 - Shading the region that satisfies all inequalities

After graphing all three lines, the region that satisfies all three inequalities is shaded in. This region is where \(x > -2\), \(y \geq -2\), and \(y < 4\), so it's the region between the lines \(y=-2\) and \(y=4\), to the right of the line \(x=-2\).

Key Concepts

System of Linear Inequalities

When dealing with a system of linear inequalities, such as the set of equations given in our exercise, we're looking at multiple linear inequalities that we need to consider simultaneously. A linear inequality resembles a linear equation but uses inequality symbols (>, <, ≤, ≥) instead of an equals sign.

• A solution to a single linear inequality is an entire range of values, not just a single point.
• When we combine multiple inequalities, we look for where their solutions overlap—this common solution area is our primary focus.
• Systems of inequalities are common in real-life problems where constraints or limitations are present, and finding a feasible set of solutions within those constraints is necessary.

In the given exercise, we have three separate inequalities involving variables x and y. The task involves graphing this system to determine the set of points that satisfies all of the inequalities at once. The graphical solution will reveal a region of the coordinate plane where these conditions hold true.

Inequality Graphing

Graphing an inequality is a way to visualize the set of solutions that satisfy the inequality. Unlike equations, inequalities do not just form a line on a graph; they form regions. The process can be broken down into simple steps:

• Draw the corresponding equation: If the inequality is 'greater than' (>), or 'less than' (<), a dashed line is used, indicating that points on the line are not included in the solution. If the inequality includes 'greater than or equal to' (≥) or 'less than or equal to' (≤), a solid line is drawn, showing that points on the line are part of the solution.
• Shade the appropriate region: The area that satisfies the inequality is shaded. For example, if the inequality is 'greater than', you would shade above the line on the graph.
• Repeat for multiple inequalities: When graphing a system, perform these steps for each inequality and identify the overlapping shaded areas that satisfy all inequalities simultaneously.

Following the solution steps results in three lines drawn on the graph, two of which are horizontal (for the 'y' inequalities) and one vertical (for the 'x' inequality), each appropriately dashed or solid based on the inequality symbols.

Shading Regions in Algebra

Shading regions is a crucial step in graphing inequalities because it visually represents the solution set. Here's what you need to remember when performing this task:

• Evaluate Each Inequality Individually: For each inequality, determine which side of the line to shade by picking a test point not on the line and checking if it satisfies the inequality.
• Intersection of Shaded Regions: When dealing with a system of inequalities, the overlapping shaded regions from each inequality represent the solution to the system. Only the points that lie within all shaded areas simultaneously are valid solutions to the system.
• Boundaries Included or Excluded: Depending on the inequality symbol, the boundary line itself might not be part of the solution. If so, use a dashed line to indicate exclusion. A solid line indicates inclusion.

In the step-by-step solution, this methodology effectively identifies the region that satisfies all three inequalities. Remarkably, step four culminates in the critical intersection where the conditions meet, highlighting the area to the right of the vertical dashed line at x = -2, between the two horizontal lines y = -2 and y = 4, solid at y = -2 indicating inclusion and dashed at y = 4 indicating exclusion.