Question

Graph the solution set of the system of inequalities. $$\left\\{\begin{aligned} 2 x+y & \geq 2 \\ x & \leq 2 \\ y & \leq 1 \end{aligned}\right.$$

Short answer

The solution set is a region inside the triangle formed by the lines of the inequalities, including the borders due to the 'less than or equal to' and 'greater than or equal to' inequalities.

Step-by-step solution

Graphing First Inequality

The first inequality is \(2x+y \geq 2\). To graph this, you would first rearrange it to the form \(y \geq -2x + 2\). This represents all the points above the line \(y = -2x +2\), including the points on the line because of the ‘greater than or equal to’ sign.

Graphing Second Inequality

The second inequality is \(x \leq 2\). This is a vertical line and includes all the points on the line and to the left of it.

Graphing Third Inequality

The third inequality is \(y \leq 1\). This is a horizontal line and includes all the points on the line and below it.

Identifying the Solution Region

The solution region of the system of inequalities is the region that satisfies all the three given conditions and is a shared region among the three graphed inequalities.

Key concepts

Graphing Inequalities

When we talk about graphing inequalities, we're essentially visualizing the solutions to inequalities on the coordinate plane. Each inequality in a system can be graphed as a line or curve, representing a boundary. Below are some key points to keep in mind:

• **Convert Inequality to Equation:** Start by treating the inequality as an equation to find its boundary.
• **Plot the Boundary:** Graph the equation as a line. Use a solid line if the inequality includes one side (\( \geq \) or \( \leq \)), or a dashed line if it doesn't (\( > \) or \( < \)).
• **Determine the Region:** Shade the appropriate area that satisfies the inequality. Use a test point if necessary (often (0,0) if it's not on the line) to determine which side to shade.

For instance, in the inequality \( 2x + y \geq 2 \), we graph the line \( y = -2x + 2 \) with a solid line, then shade above it since the inequality sign is "greater than or equal to." This shows all points (solutions) that satisfy this inequality.

Solution Region

The solution region is where the magic happens! This is the shared area on the graph where all inequalities in the system overlap, meaning all conditions are satisfied. Here's how to identify it:

• **Overlap of Shaded Regions:** After graphing all individual inequalities, observe where their shaded areas intersect. This overlapping area is your solution region.
• **Check Boundary Characteristics:** Different boundaries can impact the shape of the overlap. Vertical, horizontal, and slanted lines together form a polygon that contains or excludes certain regions.
• **Consider Each Line:** Ensure each edge of the polygon matches the correct aspect of each inequality (solid vs. dashed).

For the given exercise, the solution region will be inside where \( x \leq 2 \) and \( y \leq 1 \), above \( y = -2x + 2 \). This creates a defined area on the graph that represents all solutions satisfying all inequalities.

Algebraic Inequalities

Algebraic inequalities involve expressions containing inequality symbols \(<\), \(>\), \(\leq\), and \(\geq\). These are pivotal in determining which part of the plane will represent the solutions, but there's more to consider:

• **Solving Inequalities:** Often involves isolating the variable much like solving equations, but remember: flipping the inequality sign occurs when multiplying or dividing by a negative number.
• **System Solving:** Sometimes easier by considering the qualitative behavior of each inequality when viewed graphically—they literally outline possible solution zones.
• **Real-Life Application:** These systems model scenarios involving constraints, such as optimization problems where resources are limited.

In our system, each inequality such as \( 2x+y \geq 2 \) sets a rule for the boundary of the region. Each inequality adds another constraint, narrowing the possible solutions further to a common solution region. This graphical representation makes complex relationships between constraints more tangible.