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

Verified step by step guidance

1. Step 1: Identify the system of inequalities given: \[\left\{ \begin{array}{l} 2x - 3y \leq 6 \\ x \geq 5 \end{array} \right.\]
2. Step 2: Graph the boundary line for the first inequality, which is the line \[2x - 3y = 6\]. To do this, find the intercepts: - For the x-intercept, set \[y = 0\] and solve for \[x\]. - For the y-intercept, set \[x = 0\] and solve for \[y\].
3. Step 3: Determine which side of the line \[2x - 3y = 6\] satisfies the inequality \[2x - 3y \leq 6\]. You can test a point not on the line, such as the origin \[(0,0)\], by substituting into the inequality.
4. Step 4: Graph the vertical line \[x = 5\], which is the boundary for the second inequality \[x \geq 5\]. This line divides the plane into two regions: one where \[x\] is greater than or equal to 5, and one where \[x\] is less than 5.
5. Step 5: The solution set to the system is the region where the shaded areas from both inequalities overlap. Shade the region that satisfies both \[2x - 3y \leq 6\] and \[x \geq 5\].

Key Concepts

Graphing Linear Inequalities

Graphing linear inequalities involves shading the region of the coordinate plane that satisfies the inequality. The boundary line is drawn using the related linear equation, and it is solid if the inequality includes equality (≤ or ≥). The shaded area represents all points that make the inequality true.

System of Inequalities

A system of inequalities consists of two or more inequalities considered together. The solution set is the intersection of the individual solution regions, meaning only points that satisfy all inequalities simultaneously are included.

Interpreting Boundary Lines and Inequality Signs

The inequality sign determines which side of the boundary line to shade. For example, 'x ≥ 5' means shading all points to the right of the vertical line x = 5, including the line itself. Understanding this helps correctly identify the feasible region.