Axl.com/math/algebra-1/checkpoint-modeling-with-linear-equations-and-inequalities?lmsplatform=1 English paper takes \( \frac{1}{4} \) of an hour to grade, and each math test takes \( \frac{1}{6} \) of an hour to grade. The teacher wants to spend over 3 hours but under 6 hours on grading this weekend. 4)) Graph a system of inequalities that represents this story. One inequality should represent the minimum number of hours the teacher wants to spend grading, and the other inequality should represent the maximum number of hours she wants to spend grading, \( \begin{array}{l}\text { ()) Select a line to change it between solid and dotted. Select a region to shade it. } \\ \text { 4)) Grading over the weekend }\end{array} \)

To model the teacher's grading time constraints with a system of inequalities, follow these steps: ### **1. Define the Variables** - Let \( E \) represent the **number of English papers** to grade. - Let \( M \) represent the **number of Math tests** to grade. - Let \( T \) represent the **total time** spent grading in hours. ### **2. Express the Total Time** Each English paper takes \( \frac{1}{4} \) hour to grade, and each Math test takes \( \frac{1}{6} \) hour to grade. Therefore, the total time \( T \) spent grading is: \[ T = \frac{1}{4}E + \frac{1}{6}M \] ### **3. Set Up the Inequalities** The teacher wants to spend **more than 3 hours** but **less than 6 hours** grading. This gives us two inequalities: \[ \frac{1}{4}E + \frac{1}{6}M > 3 \quad \text{(Minimum time)} \] \[ \frac{1}{4}E + \frac{1}{6}M < 6 \quad \text{(Maximum time)} \] ### **4. Graphing the System of Inequalities** To graph this system, follow these guidelines: 1. **Rearrange the Inequalities into Slope-Intercept Form (\( M = mE + b \))** for easier graphing. - **First Inequality (\( T > 3 \)):** \[ \frac{1}{4}E + \frac{1}{6}M > 3 \] Multiply all terms by 12 to eliminate fractions: \[ 3E + 2M > 36 \\ 2M > -3E + 36 \\ M > -\frac{3}{2}E + 18 \] - **Second Inequality (\( T < 6 \)):** \[ \frac{1}{4}E + \frac{1}{6}M < 6 \] Multiply all terms by 12: \[ 3E + 2M < 72 \\ 2M < -3E + 72 \\ M < -\frac{3}{2}E + 36 \] 2. **Draw the Boundary Lines:** - **For \( M > -\frac{3}{2}E + 18 \):** - Draw the line \( M = -\frac{3}{2}E + 18 \) as a **dashed line** (since the inequality is strict: \( > \)). - **For \( M < -\frac{3}{2}E + 36 \):** - Draw the line \( M = -\frac{3}{2}E + 36 \) as a **dashed line** (since the inequality is strict: \( < \)). 3. **Shade the Feasible Region:** - The feasible region is **between** the two dashed lines. - Since both inequalities are strict, do **not** include the boundary lines themselves in the shaded area. - Ensure that \( E \) and \( M \) are non-negative, so the feasible region will be in the first quadrant. ### **5. Visual Representation** Here's a conceptual illustration of the graph:  - **Dashed Lines:** Represent the boundaries \( M = -\frac{3}{2}E + 18 \) and \( M = -\frac{3}{2}E + 36 \). - **Shaded Area:** The region between the two dashed lines where \( 3 < T < 6 \). - **Axes:** - **Horizontal (E-axis):** Number of English papers. - **Vertical (M-axis):** Number of Math tests. ### **6. Summary of the System of Inequalities** \[ \begin{cases} \frac{1}{4}E + \frac{1}{6}M > 3 \\ \frac{1}{4}E + \frac{1}{6}M < 6 \\ E \geq 0 \\ M \geq 0 \end{cases} \] ### **7. Additional Notes** - **Solid vs. Dashed Lines:** Use **dashed lines** because the inequalities are strict (they do not include equality). If the inequalities were \( \geq \) or \( \leq \), you would use solid lines. - **Shading:** Shade the region **between** the two dashed lines to represent all combinations of \( E \) and \( M \) that satisfy both inequalities. By graphing this system, you can visualize all possible numbers of English papers and Math tests the teacher can grade within the desired time frame of over 3 hours but under 6 hours.