Let's solve the system of linear inequalities step by step and determine if its solution set is **bounded** or **unbounded**. ### **Given Inequalities** 1. \( 3x - 3y \leq 24 \) 2. \( -3x - 2y \leq -14 \) 3. \( x > -3 \) 4. \( y < 2 \) --- ## **Step 1: Simplify the Inequalities** ### 1. \( 3x - 3y \leq 24 \) Divide both sides by 3: \[ x - y \leq 8 \implies y \geq x - 8 \] ### 2. \( -3x - 2y \leq -14 \) Multiply both sides by -1 (reverse the inequality): \[ 3x + 2y \geq 14 \implies 2y \geq 14 - 3x \implies y \geq 7 - \frac{3}{2}x \] ### 3. \( x > -3 \) No change needed. ### 4. \( y < 2 \) No change needed. --- ## **Step 2: Graph Each Inequality** 1. \( y \geq x - 8 \) — Region **above** the line \( y = x - 8 \). 2. \( y \geq 7 - \frac{3}{2}x \) — Region **above** the line \( y = 7 - \frac{3}{2}x \). 3. \( x > -3 \) — Region **to the right** of the vertical line \( x = -3 \). 4. \( y < 2 \) — Region **below** the horizontal line \( y = 2 \). --- ## **Step 3: Analyze the Feasible Region** - The region is above both \( y = x - 8 \) and \( y = 7 - \frac{3}{2}x \). - The region is to the right of \( x = -3 \). - The region is below \( y = 2 \). ### **Intersection and Boundary** Let's check if the region is **bounded** or **unbounded**: - The lines \( y \geq x - 8 \) and \( y \geq 7 - \frac{3}{2}x \) open **upwards** (since \( y \geq \)), and \( x > -3 \) is just a vertical boundary. - However, \( y < 2 \) puts a cap at the top. - The region is **bounded below** by the intersection of the two lines, on the left by the vertical line, and **above** by \( y = 2 \). ### **But**: Both \( y \geq x - 8 \) and \( y \geq 7 - \frac{3}{2}x \) are **unbounded upwards** unless restricted by \( y < 2 \), which is present here. The intersection of all these half-planes forms a **bounded quadrilateral region**. There is an upper limit (\( y < 2 \)), a lower limit (where the two lines intersect), and a left boundary (\( x > -3 \)). So, **the region is bounded**. --- ## **Step 4: Final Answer** - **The solution set is bounded.** --- ## **Summary Table** | Inequality | Region (in terms of y) | Region Description | |----------------------------|------------------------------------|-----------------------------------| | \( x - y \leq 8 \) | \( y \geq x - 8 \) | Above \( y = x - 8 \) | | \( -3x - 2y \leq -14 \) | \( y \geq 7 - \frac{3}{2}x \) | Above \( y = 7 - \frac{3}{2}x \) | | \( x > -3 \) | | Right of \( x = -3 \) | | \( y < 2 \) | | Below \( y = 2 \) | --- ### **Final Answer** > **The solution set is bounded.** If you need the graph or further explanation on intersection points, let me know!

## Graph of the System of Inequalities To visualize the solution set for the given inequalities, we can sketch the lines and shade the appropriate regions. ### **Inequalities to Graph** 1. **Line for** \( y = x - 8 \) - Slope: 1 - Y-intercept: -8 2. **Line for** \( y = 7 - \frac{3}{2}x \) - Slope: -1.5 - Y-intercept: 7 3. **Vertical line** at \( x = -3 \) 4. **Horizontal line** at \( y = 2 \) ### **Steps to Create the Graph** 1. **Draw the lines**: - For \( y = x - 8 \), plot points like (-8, -8) and (8, 0) and connect them. - For \( y = 7 - \frac{3}{2}x \), plot points like (0, 7) and (4, -1) and connect them. - Draw a dashed vertical line at \( x = -3 \) (since \( x > -3 \)). - Draw a dashed horizontal line at \( y = 2 \) (since \( y < 2 \)). 2. **Shade the regions**: - Shade **above** the line \( y = x - 8 \). - Shade **above** the line \( y = 7 - \frac{3}{2}x \). - Shade **to the right** of the line \( x = -3 \). - Shade **below** the line \( y = 2 \). ### **Intersection Points** To find the intersection points of the two lines, set them equal to each other: \[ x - 8 = 7 - \frac{3}{2}x \] ### **Solving for Intersection Points** 1. Rearranging gives: \[ x + \frac{3}{2}x = 15 \] \[ \frac{5}{2}x = 15 \implies x = 6 \] 2. Substitute \( x = 6 \) back into either equation to find \( y \): \[ y = 6 - 8 = -2 \] ### **Final Graph Illustration** - The bounded region will be formed between the lines and the horizontal and vertical boundaries. Here is a sketch of the graph based on the inequalities: ```plaintext | 2|---------------------- (y = 2) | | | | -+---------------------------- (x = -3) -8| * (6, -2) | | -3|---------------------- (y = x - 8) | | | | | -14|------------------- (y = 7 - (3/2)x) | ``` ### **Conclusion** - The shaded area represents the feasible region, confirming that the solution set is indeed bounded. If you need a more accurate graphical representation, consider using graphing software to plot these inequalities.