Problem 39 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The solution set of the system of inequalities is the region above the line of equation \(y = x - 1\) and to the right of the line \(x = 2\).

Step 1: Analyze and plot the inequalities

Start by analyzing and converting the inequalities to the line equation format (y = mx + b). The first inequality \(x - y \leq 1\) can be rewritten as \(y \geq x - 1\). The second inequality \(x \geq 2\) will be represented by a vertical line. Plot both the lines on the graph.

Step 2: Shade the appropriate region

Now, based on the sign of the inequalities, shade the regions. For the first inequality \(y \geq x - 1\), shade above the line. For the second inequality \(x \geq 2\), shade to the right of the line.

Step 3: Identify the solution set

The solution to the system of inequalities is the region where the shading overlaps. This region satisfies both inequalities and thus, represents the solution set of our system.

Key Concepts

graphing inequalities

Graphing inequalities is a crucial part of visualizing solutions to problems involving relationships and comparisons between mathematical expressions. It transforms an inequality into a graphical representation on a coordinate plane, which makes it easier to identify the range of possible solutions. The process involves several key steps:

• Convert inequalities to equations: Change the inequality sign to an equality to find the line which will guide the understanding of bounds.
• Plot the inequality boundary: Graph the line. For example, the inequality \(x - y \leq 1\) becomes the line \(y = x - 1\).
• Shade the solution region: Determine which side of the line to shade. The inequality \(y \geq x - 1\) indicates shading above the line. Similarly, for \(x \geq 2\), we shade to the right of the vertical line at \(x = 2\).

By plotting these elements, one can visually assess the solution set of the inequalities. The key takeaway is that the intersection of shaded areas reveals the comprehensive solution of the system.

solution set

The solution set in a system of inequalities is the collection of values for which all inequalities in the system hold true simultaneously. It is essentially the common region where all shaded areas from each inequality overlap.

• Graphically, after plotting each inequality, the overlapping shaded area illustrates where each inequality's conditions are satisfied.
• For our example, the solution set is where the shaded region for \(y \geq x - 1\) and the shaded region for \(x \geq 2\) intersect.
• This overlapping region represents all ordered pairs \((x, y)\) that make all the inequalities in the system true.

Identifying the solution set is vital because it offers a visual representation that confirms whether a system of inequalities possesses solutions and if so, where they are located on the coordinate plane.

linear inequalities

Linear inequalities are inequalities involving linear expressions. Much like linear equations, they include variables raised only to the first power, but instead of an equal sign, they have inequality symbols like \(>\), \(<\), \(\geq\), or \(\leq\).

• A typical linear inequality in two variables might resemble \(y \geq mx + b\), which describes a region in the coordinate plane, rather than a line.
• To resolve these, you'll often compare your expressions directly or rearrange them into a more familiar line form, \(y = mx + b\). For instance, if starting with \(x - y \leq 1\), you rearrange to \(y \geq x - 1\).

The simplicity of linear inequalities translates into straightforward graphing and solution techniques. As you work with them, you gain insights into the relationships between different variables under the specified constraints.