Consider the following system of linear inequalities and corresponding graph: \( \begin{array}{l}y \geq 2 x-4 \\ x>-3 \\ y<5\end{array} \)

Certainly! Let's analyze and graph the given system of linear inequalities: \[ \begin{cases} y \geq 2x - 4 \\ x > -3 \\ y < 5 \end{cases} \] ### Step 1: Graph the Boundary Lines 1. **First Inequality: \( y \geq 2x - 4 \)** - **Boundary Line:** \( y = 2x - 4 \) (a **solid** line because the inequality is \(\geq\)) - **Slope (m):** 2 - **Y-intercept (b):** \(-4\) 2. **Second Inequality: \( x > -3 \)** - **Boundary Line:** \( x = -3 \) (a **dashed** vertical line because the inequality is \(>\)) 3. **Third Inequality: \( y < 5 \)** - **Boundary Line:** \( y = 5 \) (a **dashed** horizontal line because the inequality is \(<\)) ### Step 2: Plot Each Boundary Line 1. **Plotting \( y = 2x - 4 \):** - Start at \( (0, -4) \) on the y-axis. - Use the slope to find another point: from \( (0, -4) \), rise 2 units and run 1 unit to the right to reach \( (1, -2) \). - Draw a **solid** line through these points. 2. **Plotting \( x = -3 \):** - Draw a **dashed** vertical line crossing the x-axis at \(-3\). 3. **Plotting \( y = 5 \):** - Draw a **dashed** horizontal line crossing the y-axis at \(5\). ### Step 3: Determine the Shaded Regions 1. **For \( y \geq 2x - 4 \):** - Shade **above** the line \( y = 2x - 4 \). 2. **For \( x > -3 \):** - Shade **to the right** of the line \( x = -3 \). 3. **For \( y < 5 \):** - Shade **below** the line \( y = 5 \). ### Step 4: Identify the Feasible Region The **feasible region** is where all shaded areas overlap. This region satisfies all three inequalities simultaneously. ### Step 5: Find Intersection Points (Vertices) To better understand the feasible region, let's find where the boundary lines intersect: 1. **Intersection of \( y = 2x - 4 \) and \( y = 5 \):** \[ 2x - 4 = 5 \\ 2x = 9 \\ x = 4.5 \\ \Rightarrow (4.5, 5) \] 2. **Intersection of \( y = 2x - 4 \) and \( x = -3 \):** \[ y = 2(-3) - 4 = -6 - 4 = -10 \\ \Rightarrow (-3, -10) \] 3. **Intersection of \( y = 5 \) and \( x = -3 \):** \[ \Rightarrow (-3, 5) \] ### Step 6: Sketch the Graph While I can't provide a visual plot here, here's a description to help you sketch it: 1. **Draw the axes** with appropriate scales. 2. **Plot the boundary lines**: - Draw a solid line for \( y = 2x - 4 \). - Draw a dashed vertical line at \( x = -3 \). - Draw a dashed horizontal line at \( y = 5 \). 3. **Shade the regions**: - Shade above the solid line \( y = 2x - 4 \). - Shade to the right of the dashed line \( x = -3 \). - Shade below the dashed line \( y = 5 \). 4. **Identify the overlapping area**, which is bounded: - **On the left** by \( x = -3 \). - **Below** by \( y = 2x - 4 \). - **Above** by \( y = 5 \). This overlapping (feasible) region represents all the points \((x, y)\) that satisfy all three inequalities simultaneously. ### Step 7: Analyze the Feasible Region - **Boundaries:** - The vertical boundary \( x = -3 \) limits the region to \( x > -3 \). - The line \( y = 2x - 4 \) serves as the lower boundary. - The horizontal line \( y = 5 \) serves as the upper boundary. - **Vertices (Corner Points):** - Intersection of \( y = 2x - 4 \) and \( x = -3 \): \((-3, -10)\) - Intersection of \( y = 2x - 4 \) and \( y = 5 \): \((4.5, 5)\) - Intersection of \( y = 5 \) and \( x = -3 \): \((-3, 5)\) These vertices can be useful for optimization problems or further analysis within the feasible region. ### Summary The solution to the system of inequalities is the set of all points \((x, y)\) that lie: - **To the right** of \( x = -3 \). - **Above** the line \( y = 2x - 4 \). - **Below** the line \( y = 5 \). This region is a trapezoidal area on the coordinate plane bounded by the lines mentioned above.