To graph the linear inequality [tex]\(\frac{1}{2} x - 2 y > -6\)[/tex], follow these step-by-step instructions:
1. Convert the Inequality to an Equation:
We begin by turning the inequality into an equation to find its boundary line:
[tex]\[
\frac{1}{2} x - 2 y = -6
\][/tex]
2. Clear the Fraction:
To eliminate the fraction, multiply every term in the equation by 2:
[tex]\[
x - 4 y = -12
\][/tex]
3. Solve for [tex]\(y\)[/tex]:
Rearrange the equation into the slope-intercept form [tex]\(y = mx + b\)[/tex]:
[tex]\[
x - 4y = -12 \implies -4y = -x - 12 \implies y = \frac{1}{4}x + 3
\][/tex]
Here, the slope [tex]\(m = \frac{1}{4}\)[/tex] and the y-intercept [tex]\(b = 3\)[/tex].
4. Plot the Boundary Line:
Plot the boundary line [tex]\(y = \frac{1}{4}x + 3\)[/tex].
- At [tex]\(x = 0\)[/tex]:
[tex]\[
y = \frac{1}{4}(0) + 3 = 3 \quad \text{(Point: (0, 3))}
\][/tex]
- At [tex]\(x = 4\)[/tex]:
[tex]\[
y = \frac{1}{4}(4) + 3 = 1 + 3 = 4 \quad \text{(Point: (4, 4))}
\][/tex]
Plot the points [tex]\((0, 3)\)[/tex] and [tex]\((4, 4)\)[/tex] and draw a straight line through them.
5. Determine the Inequality Region:
Since our original inequality is [tex]\(\frac{1}{2} x - 2 y > -6\)[/tex], we need to determine which side of the boundary line satisfies this inequality.
- Choose a test point that is not on the boundary line (e.g., the origin [tex]\((0,0))\)[/tex]):
[tex]\[
\frac{1}{2}(0) - 2(0) = 0
\][/tex]
Since [tex]\(0 > -6\)[/tex] is true, the region containing [tex]\((0,0)\)[/tex] is the solution.
6. Shade the Correct Region:
Shade the region above the boundary line [tex]\(y = \frac{1}{4}x + 3\)[/tex], as it satisfies the inequality [tex]\(\frac{1}{2} x - 2 y > -6\)[/tex]. Since the inequality is strictly greater than ([tex]\(>\)[/tex]), the boundary line itself is not included in the solution. Use a dashed line to represent that the boundary line is not part of the solution.
The final graph displays a dashed line for [tex]\(y = \frac{1}{4}x + 3\)[/tex] and a shaded region above this line to represent all the [tex]\( (x,y) \)[/tex] pairs that satisfy the inequality [tex]\(\frac{1}{2} x - 2 y > -6\)[/tex].
