In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. 2x−5y≤10, 3x−2y>6

Verified step by step guidance

1. Step 1: Identify each inequality and rewrite them in slope-intercept form (y = mx + b) to make graphing easier. For the first inequality, \( x - 4y \leq 8 \), solve for \( y \):
2. \[ x - 4y \leq 8 \implies -4y \leq 8 - x \implies y \geq \frac{x - 8}{4} \] (Note the inequality direction changes when dividing by a negative number.)
3. Step 2: For the second inequality, \( 4x - 2y > 12 \), solve for \( y \):
4. \[ 4x - 2y > 12 \implies -2y > 12 - 4x \implies y < \frac{4x - 12}{2} = 2x - 6 \] (Again, inequality direction changes when dividing by a negative number.)
5. Step 3: Graph the boundary lines \( y = \frac{x - 8}{4} \) and \( y = 2x - 6 \). Use a solid line for \( y = \frac{x - 8}{4} \) because the inequality is \( \leq \), and a dashed line for \( y = 2x - 6 \) because the inequality is strict (\( > \) or \( < \)).
6. Step 4: Determine the solution region for each inequality by testing a point not on the boundary lines (commonly the origin \( (0,0) \) if it is not on the line). For \( y \geq \frac{x - 8}{4} \), check if \( (0,0) \) satisfies the inequality. For \( y < 2x - 6 \), do the same.
7. Step 5: The solution set of the system is the intersection of the two regions found in Step 4. Shade the overlapping region on the graph to represent all points that satisfy both inequalities simultaneously.

Key Concepts

Graphing Linear Inequalities

Graphing linear inequalities involves plotting the boundary line represented by the related linear equation and then shading the region that satisfies the inequality. For '≤' or '≥', the boundary line is solid, indicating points on the line are included. For '<' or '>', the line is dashed, showing points on the line are excluded.

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. Graphing the system helps visualize where these regions overlap, representing all possible solutions.

Testing Points to Determine Solution Regions

To identify which side of the boundary line to shade, select a test point not on the line (commonly the origin if not on the line) and substitute it into the inequality. If the inequality holds true, shade the side containing the test point; otherwise, shade the opposite side. This method ensures accurate graphing of solution regions.