Problem 35 - Algebra and Trigonometry

Question

Graph the solution set of each system of inequalities or indicate that the system has no solution. $$ \left\\{\begin{array}{l} {x \leq 2} \\ {y \geq-1} \end{array}\right. $$

Step-by-Step Solution

Answer

The solution to the given inequalities is a rectangular region which is to the left of the line \(x=2\) (including the line) and above the line \(y=-1\) (including the line).

Step 1: Graphing the inequality \(x \leq 2\)

To graph the inequality \(x \leq 2\), draw a vertical line at \(x=2\). Since the inequality is less than or equal to, all the points to the left of this line (including the line itself) will be part of this solution.

Step 2: Graphing the inequality \(y \geq -1\)

Next, to graph the inequality \(y \geq -1\), draw a horizontal line at \(y=-1\). Since the inequality is greater than or equal to, all the points above this line (including the line itself) will be part of this solution.

Step 3: Finding the solution of the system

Having both inequalities drawn, search for an area that is part of both solutions. This will be an area above \(y=-1\) line (including the line) and to the left of \(x=2\) line (including the line). This is the common solution area for the system of inequalities.

Key Concepts

Graphing Inequalities

Graphing inequalities involves representing the set of solutions for an inequality on a coordinate plane. Start by understanding the inequality. If it involves \(x\), it's related to vertical lines, while \(y\) relates to horizontal ones. For example, the inequality \(x \leq 2\) means the graph will include a solid vertical line at \(x = 2\). This line shows that \(x\) can be exactly 2, and it shades to the left because \(x\) can also be less than 2. Likewise, for \(y \geq -1\), you would draw a solid horizontal line at \(y = -1\) and shade upwards. The line indicates \(y = -1\) is included, with shading upwards showing \(y\) can be greater as well. Graphing this way visually represents multiple possible solutions, helping in understanding and solving systems of inequalities.

Solution Sets

Solution sets for a system of inequalities encompass all the possible points satisfying all inequalities in the system. When understanding systems of inequalities, think of them as condition groups. You need solutions that meet all conditions simultaneously. After graphing each inequality separately, you identify the overlapping region. This intersection represents the solution set. It may be a region of the coordinate plane where multiple shaded areas overlap. For systems like \(x \leq 2\) and \(y \geq -1\), the solution set is where the conditions for both inequalities are true — to the left of \(x = 2\) and above \(y = -1\). This area is part of both graphical representations. It includes points that lie on the border lines as well.

Two-variable Inequalities

Two-variable inequalities involve expressions with two different variables, often \(x\) and \(y\). Solving them means finding all \((x, y)\) pairs satisfying these conditions. These appear often in questions as systems, pairing various inequalities together. In systems like these, each inequality affects the other because they define a combined solution space. Here, for example, with both \(x \leq 2\) and \(y \geq -1\), the solution isn't just a single point or line; it encompasses an area where both conditions are met. Understanding two-variable inequalities is crucial to handling more complex algebraic problems. By mastering graphing and working with these conditions simultaneously, students gain the ability to tackle and visualize higher-level algebraic challenges.