Alright, let's solve the inequality [tex]\( 3x - 2y < 10 \)[/tex].
Step-by-Step Solution:
1. Understanding the Inequality:
- We have an inequality involving two variables, [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
2. Isolate [tex]\( y \)[/tex] on one side (Optional for graphing purposes):
- While it's not mandatory, isolating [tex]\( y \)[/tex] on one side can make it easier to understand and graph the inequality.
- Let's rearrange [tex]\( 3x - 2y < 10 \)[/tex] to solve for [tex]\( y \)[/tex].
[tex]\[
3x - 2y < 10
\][/tex]
Subtract [tex]\( 3x \)[/tex] from both sides:
[tex]\[
-2y < 10 - 3x
\][/tex]
Divide both sides by [tex]\(-2\)[/tex] and reverse the inequality sign (since we are dividing by a negative number):
[tex]\[
y > \frac{10 - 3x}{-2}
\][/tex]
Simplify the fraction:
[tex]\[
y > -\frac{10}{2} + \frac{3x}{2}
\][/tex]
[tex]\[
y > -5 + \frac{3}{2}x
\][/tex]
So, the inequality can also be written as:
[tex]\[
y > \frac{3}{2}x - 5
\][/tex]
3. Graphical Interpretation:
- To graph [tex]\( 3x - 2y < 10 \)[/tex], first graph the boundary line [tex]\( 3x - 2y = 10 \)[/tex].
- Rewrite the equality [tex]\( 3x - 2y = 10 \)[/tex] in slope-intercept form [tex]\( y = mx + b \)[/tex] for easier graphing:
[tex]\[
3x - 2y = 10 \rightarrow -2y = -3x + 10 \rightarrow y = \frac{3}{2}x - 5
\][/tex]
- The line [tex]\( y = \frac{3}{2}x - 5 \)[/tex] has a slope of [tex]\( \frac{3}{2} \)[/tex] and a y-intercept at [tex]\( (0, -5) \)[/tex].
4. Shading the Appropriate Region:
- Since the inequality is [tex]\( 3x - 2y < 10 \)[/tex], we need to determine which side of the line [tex]\( y = \frac{3}{2}x - 5 \)[/tex] to shade.
- We shade the region where [tex]\( y \)[/tex] is greater than [tex]\( \frac{3}{2}x - 5 \)[/tex]. This means we shade above the line.
By visualizing the inequality on a graph and understanding the algebraic manipulation, we interpret the inequality [tex]\( 3x - 2y < 10 \)[/tex] effectively.
