To graph the linear inequality [tex]\(6x + 2y > -10\)[/tex], follow these steps:
1. Rewrite the Inequality:
First, let's rewrite the inequality in a more standard form, similar to the equation of a line but with an inequality sign:
[tex]\[ 6x + 2y > -10 \][/tex]
2. Convert to Equality for Boundary Line:
Convert the inequality to an equality to find the boundary line:
[tex]\[ 6x + 2y = -10 \][/tex]
3. Find Intercepts for the Boundary Line:
- Y-intercept: Set [tex]\(x = 0\)[/tex]:
[tex]\[ 6(0) + 2y = -10 \][/tex]
[tex]\[ 2y = -10 \][/tex]
[tex]\[ y = -5 \][/tex]
- X-intercept: Set [tex]\(y = 0\)[/tex]:
[tex]\[ 6x + 2(0) = -10 \][/tex]
[tex]\[ 6x = -10 \][/tex]
[tex]\[ x = -\frac{10}{6} = -\frac{5}{3} \approx -1.67 \][/tex]
4. Plot the Boundary Line:
Plot the boundary line [tex]\(6x + 2y = -10\)[/tex] using the intercepts:
- Point (0, -5)
- Point [tex]\((-1.67, 0)\)[/tex]
Draw a dashed line (because the inequality is strict and does not include the line itself).
5. Determine Which Side of the Line to Shade:
To figure out which side of the line to shade, select a test point that is not on the line. Commonly, the origin (0, 0) is a convenient choice:
[tex]\[ 6(0) + 2(0) > -10 \][/tex]
[tex]\[ 0 > -10 \][/tex]
This statement is true, indicating that the origin satisfies the inequality. Therefore, the region containing the origin is the solution region.
6. Shade the Solution Region:
Shade the region above the dashed line [tex]\(6x + 2y = -10\)[/tex], which represents all [tex]\((x, y)\)[/tex] pairs that satisfy [tex]\(6x + 2y > -10\)[/tex].
Final Graph Summary:
- Boundary Line: [tex]\(6x + 2y = -10\)[/tex], represented by a dashed line.
- Intercepts: (0, -5) and [tex]\((-1.67, 0)\)[/tex]
- Test Point: (0, 0), it satisfies the inequality.
- Shaded Region: Above the boundary line [tex]\(6x + 2y = -10\)[/tex].
This is the graph representing the given linear inequality [tex]\(6x + 2y > -10\)[/tex].
