Grapht $$ \left\{\begin{array}{l}y-x+2\end{array} ight. $$

Question

Grapht $$ \left\{\begin{array}{l}y<4 x+1 \ y>-x+2\end{array}ight. $$

UpStudy AI · Accepted Answer

We wish to graph the system y < 4x + 1 y > –x + 2 follow these steps: 1. Find the boundary lines: • Line 1: y = 4x + 1 • Line 2: y = –x + 2 Note: Since the inequalities are strict (< and >), these lines themselves are not part of the solution region. When you draw them, use dashed lines. 2. Find the intersection point of the two lines: Set 4x + 1 = –x + 2 4x + x = 2 – 1 5x = 1 x = 1/5 Then substitute back to find y: y = 4(1/5) + 1 = 4/5 + 1 = 9/5 The lines intersect at (1/5, 9/5). 3. Decide which side of each line to shade. • For y < 4x + 1: Pick a test point not on the line (0,0) is convenient. Test: 0 < 4(0) + 1 ⟹ 0 < 1, which is true. Thus, the region containing (0,0) is the solution for y < 4x + 1. • For y > –x + 2: Again, test (0,0). Test: 0 > –(0) + 2 ⟹ 0 > 2, which is false. So pick another test point; try (0,3). Test: 3 > –(0) + 2 ⟹ 3 > 2, which is true. Thus, the region containing (0,3) is the solution for y > –x + 2. 4. Shade the common (overlapping) region: • Shade below the dashed line y = 4x + 1 • Shade above the dashed line y = –x + 2 The solution to the system is the area where these two shaded regions overlap. 5. Sketching: • Draw the two dashed boundary lines. • Mark their intersection at (1/5, 9/5). • Shade the region that lies below y = 4x + 1 and above y = –x + 2. This overlapping area is your final answer. This graph represents all points (x, y) that satisfy y < 4x + 1 and y > –x + 2. To graph the system $$ y < 4x + 1 $$ and $$ y > -x + 2 $$, follow these steps: 1. **Draw the boundary lines**: - $$ y = 4x + 1 $$ (dashed line) - $$ y = -x + 2 $$ (dashed line) 2. **Find the intersection point**: - Solve $$ 4x + 1 = -x + 2 $$ to get $$ x = \frac{1}{5} $$ and $$ y = \frac{9}{5} $$. 3. **Determine the shading**: - For $$ y < 4x + 1 $$, shade below the line. - For $$ y > -x + 2 $$, shade above the line. 4. **Identify the solution region**: - The overlapping shaded area represents all points that satisfy both inequalities. This graph shows all points (x, y) that meet both $$ y < 4x + 1 $$ and $$ y > -x + 2 $$.

Grapht \( \left\{\begin{array}{l}y<4 x+1 \\ y>-x+2\end{array}\right. \)

Upstudy AI Solution

Answer

Solution

Extra Insights

Related Questions

Latest Algebra Questions