In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. x+y>4, x+y<−1

Verified step by step guidance

1. Step 1: Identify the system of inequalities: \( \begin{cases} -2x + y > 6 \\ -2x + y < -4 \end{cases} \). These represent two linear inequalities with the same left-hand side but different right-hand sides.
2. Step 2: Rewrite each inequality in slope-intercept form (\( y = mx + b \)) to better understand the boundary lines. For the first inequality, add \( 2x \) to both sides to get \( y > 2x + 6 \). For the second inequality, similarly, \( y < 2x - 4 \).
3. Step 3: Graph the boundary lines \( y = 2x + 6 \) and \( y = 2x - 4 \). Since the inequalities are strict (\( > \) and \( < \)), use dashed lines to indicate that points on the lines are not included in the solution.
4. Step 4: Determine the solution region for each inequality. For \( y > 2x + 6 \), shade the region above the line \( y = 2x + 6 \). For \( y < 2x - 4 \), shade the region below the line \( y = 2x - 4 \).
5. Step 5: Identify the intersection of the two shaded regions. Since the first region is above a line that is always higher than the second line (because \( 2x + 6 > 2x - 4 \) for all \( x \)), there is no overlap between the two solution sets. Therefore, the system has no solution.

Key Concepts

Graphing Linear Inequalities

Graphing linear inequalities involves plotting the boundary line represented 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 that satisfy each inequality individually. If no common region exists, the system has no solution. Graphing helps visualize this intersection or lack thereof.

Slope-Intercept Form and Boundary Lines

Rearranging inequalities into slope-intercept form (y = mx + b) makes graphing easier by identifying the slope and y-intercept of the boundary line. For example, from -2x + y > 6, rewrite as y > 2x + 6. This form helps plot the line and determine which side to shade based on the inequality.