To understand the graph of the linear inequality [tex]\(2x - 3y < 12\)[/tex], let's break down the solution into manageable steps.
1. Start by finding the boundary line: The inequality [tex]\(2x - 3y < 12\)[/tex] can be viewed as having a boundary given by the equation [tex]\(2x - 3y = 12\)[/tex]. This is because the inequality becomes an equality at the boundary line.
2. Rewrite the boundary line in standard form: To graph this line easily, we prefer to express [tex]\(y\)[/tex] in terms of [tex]\(x\)[/tex]:
[tex]\[
2x - 3y = 12 \Rightarrow 3y = 2x - 12 \Rightarrow y = \frac{2}{3}x - 4
\][/tex]
Thus, the equation of the boundary line is [tex]\(y = \frac{2}{3}x - 4\)[/tex].
3. Interpret the boundary line: The line [tex]\(y = \frac{2}{3}x - 4\)[/tex] will serve as the boundary for the inequality. It is a straight line with a slope of [tex]\(\frac{2}{3}\)[/tex] and a y-intercept of [tex]\(-4\)[/tex].
4. Determine the inequality region: The inequality [tex]\(2x - 3y < 12\)[/tex] tells us that we are interested in the region below this boundary line. This is because replacing the inequality with [tex]\(<\)[/tex] means we’re looking for the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that make the expression [tex]\(2x - 3y\)[/tex] less than 12.
5. Graphical representation:
- Draw the boundary line [tex]\(y = \frac{2}{3}x - 4\)[/tex] on a coordinate plane.
- Since the inequality is strict (using "<" instead of "≤"), the boundary line itself is not included in the solution set. Thus, the line should be drawn as a dashed line.
- Shade the region below the dashed line to represent all the points that satisfy [tex]\(2x - 3y < 12\)[/tex].
6. Conclusion: The graph of the linear inequality [tex]\(2x - 3y < 12\)[/tex] is represented by a dashed boundary line [tex]\(y = \frac{2}{3}x - 4\)[/tex] and the region below this line. This shaded region represents all the [tex]\( (x, y) \)[/tex] pairs that satisfy the inequality.
Summary: The graph of the linear inequality [tex]\(2x - 3y < 12\)[/tex] is a dashed line for [tex]\(y = \frac{2}{3}x - 4\)[/tex] with the region below it shaded.
