Graph the solution set of each system of inequalities or indicate that the system has no solution.
$\begin{cases}x + y > 4 \\x + y > -1\end{cases}$

Verified step by step guidance

1. Start by examining each inequality separately. The system is: \[\begin{cases} 2x - y > 4 \\ 2x - y > -5 \end{cases}\]
2. Rewrite each inequality in terms of \(y\) to better understand the boundary lines: For the first inequality: \[2x - y > 4 \implies -y > 4 - 2x \implies y < 2x - 4\] For the second inequality: \[2x - y > -5 \implies -y > -5 - 2x \implies y < 2x + 5\]
3. Graph the boundary lines \(y = 2x - 4\) and \(y = 2x + 5\). Since the inequalities are strict (greater than), these lines will be dashed to indicate points on the line are not included.
4. Determine the solution region for each inequality: - For \(y < 2x - 4\), shade the region below the line \(y = 2x - 4\). - For \(y < 2x + 5\), shade the region below the line \(y = 2x + 5\).
5. The solution to the system is the intersection of these two shaded regions. Since \$2x - 4\( is always less than \)2x + 5\( for all \)x\(, the region below \)y = 2x - 4\( is contained within the region below \)y = 2x + 5\(. Therefore, the solution set is all points below the line \)y = 2x - 4$.

Key Concepts

Graphing Linear Inequalities

Graphing linear inequalities involves plotting the boundary line given by the corresponding linear equation and then shading the region that satisfies the inequality. The boundary line is dashed if the inequality is strict (>, <) and solid if it includes equality (≥, ≤). This visual representation helps identify all points that satisfy the inequality.

System of Inequalities

A system of inequalities consists of two or more inequalities considered simultaneously. The solution set is the intersection of the regions satisfying each inequality. Graphing each inequality and finding the overlapping shaded region reveals the solution set, or shows if no solution exists when regions do not overlap.

Slope-Intercept Form and Boundary Lines

Rearranging inequalities into slope-intercept form (y = mx + b) helps graph the boundary lines easily. For example, 2x - y > 4 can be rewritten as y < 2x - 4. Understanding the slope and y-intercept allows accurate drawing of the boundary line, which is essential for correctly shading the solution region.