For each of the linear inequalities, generate the associated line on the coordinate grid without a graphing calculator. To graph the lines, you will need to identify coordinates that can be represented by integers. Once the lines have been created, select the region containing the solutions to the system of linear inequalities. \( y \leq 3 \) \( x+3 y<5 \) \( y \geq 2 x-6 \)

To graph the system of linear inequalities without a graphing calculator, follow these steps for each inequality: 1. **Graph the Boundary Lines** 2. **Determine the Shading Region for Each Inequality** 3. **Identify the Solution Region Where All Shaded Areas Overlap** Let's apply these steps to each inequality in the system: ### 1. Graphing \( y \leq 3 \) **Boundary Line:** - The equality \( y = 3 \) is a **horizontal line** that passes through all points where the y-coordinate is 3. **Plotting Points:** - Choose integer values for \( x \) to find corresponding \( y \): - When \( x = -2 \), \( y = 3 \) → Point \((-2, 3)\) - When \( x = 0 \), \( y = 3 \) → Point \((0, 3)\) - When \( x = 2 \), \( y = 3 \) → Point \((2, 3)\) **Drawing the Line:** - Draw a horizontal line through the points \((-2, 3)\), \((0, 3)\), and \((2, 3)\). **Shading:** - Since the inequality is \( y \leq 3 \), shade **below** the line. ### 2. Graphing \( x + 3y < 5 \) **Boundary Line:** - Convert the inequality to the equality \( x + 3y = 5 \). **Plotting Points:** - Choose integer values for \( x \) to find corresponding \( y \): - When \( x = 0 \), \( 3y = 5 \) → \( y = \frac{5}{3} \) → Point \((0, \frac{5}{3})\) - When \( y = 0 \), \( x = 5 \) → Point \((5, 0)\) **Drawing the Line:** - Draw a straight line through \((0, \frac{5}{3})\) and \((5, 0)\). **Shading:** - Since the inequality is \( x + 3y < 5 \), shade **below** the line. ### 3. Graphing \( y \geq 2x - 6 \) **Boundary Line:** - Convert the inequality to the equality \( y = 2x - 6 \). **Plotting Points:** - Choose integer values for \( x \) to find corresponding \( y \): - When \( x = 0 \), \( y = -6 \) → Point \((0, -6)\) - When \( x = 3 \), \( y = 0 \) → Point \((3, 0)\) **Drawing the Line:** - Draw a straight line through \((0, -6)\) and \((3, 0)\). **Shading:** - Since the inequality is \( y \geq 2x - 6 \), shade **above** the line. ### 4. Identifying the Solution Region The solution to the system of inequalities is the region where all three shaded areas overlap. To precisely identify this region, follow these steps: **Finding Intersection Points:** 1. **Intersection of \( y = 3 \) and \( x + 3y = 5 \):** \[ y = 3 \\ x + 3(3) = 5 \\ x + 9 = 5 \\ x = -4 \\ \] → Intersection Point: \((-4, 3)\) 2. **Intersection of \( y = 3 \) and \( y = 2x - 6 \):** \[ 3 = 2x - 6 \\ 2x = 9 \\ x = 4.5 \\ \] → Intersection Point: \((4.5, 3)\) 3. **Intersection of \( x + 3y = 5 \) and \( y = 2x - 6 \):** \[ y = 2x - 6 \\ x + 3(2x - 6) = 5 \\ x + 6x - 18 = 5 \\ 7x = 23 \\ x = \frac{23}{7} \approx 3.2857 \\ y = 2\left(\frac{23}{7}\right) - 6 = \frac{46}{7} - \frac{42}{7} = \frac{4}{7} \approx 0.571 \\ \] → Intersection Point: \(\left(\frac{23}{7}, \frac{4}{7}\right)\) **Conclusion:** - **Solution Region:** The overlapping shaded area bounded by: - The horizontal line \( y = 3 \) (shaded below), - The line \( x + 3y = 5 \) (shaded below), and - The line \( y = 2x - 6 \) (shaded above). - **Vertices of the Solution Region:** - \((-4, 3)\) - \(\left(\frac{23}{7}, \frac{4}{7}\right)\) - \((4.5, 3)\) This region represents all the points \((x, y)\) that simultaneously satisfy all three inequalities. --- **Visual Representation:** While I can't provide a drawn graph here, you can create one by following these steps: 1. **Draw the Coordinate Axes:** - Label the x-axis and y-axis with appropriate scales. 2. **Plot Each Boundary Line:** - **\( y = 3 \):** A horizontal line crossing the y-axis at 3. - **\( x + 3y = 5 \):** A line passing through \((0, \frac{5}{3})\) and \((5, 0)\). - **\( y = 2x - 6 \):** A line passing through \((0, -6)\) and \((3, 0)\). 3. **Shade the Appropriate Regions:** - **\( y \leq 3 \):** Shade below the horizontal line. - **\( x + 3y < 5 \):** Shade below the sloping line. - **\( y \geq 2x - 6 \):** Shade above the other sloping line. 4. **Identify the Overlapping Shaded Area:** - This is your solution region, bounded by the intersection points identified above. By carefully plotting each line and shading the correct regions, you'll visually identify the set of all points that satisfy all three inequalities.