To solve the inequality [tex]\( y < -\frac{3}{5} x \)[/tex], we need to understand what the inequality represents and determine which values for [tex]\( y \)[/tex] and [tex]\( x \)[/tex] satisfy it.
1. Understand the Inequality:
The inequality [tex]\( y < -\frac{3}{5} x \)[/tex] represents a region in the coordinate plane. Specifically, it tells us that for any given value of [tex]\( x \)[/tex], the corresponding [tex]\( y \)[/tex]-value must be less than [tex]\( -\frac{3}{5} x \)[/tex] in order to satisfy the inequality.
2. Graphical Representation:
To visualize this, consider the equation [tex]\( y = -\frac{3}{5} x \)[/tex]. This equation represents a straight line with a slope of [tex]\( -\frac{3}{5} \)[/tex].
- The slope [tex]\( -\frac{3}{5} \)[/tex] means that for every increase of 1 unit in [tex]\( x \)[/tex], [tex]\( y \)[/tex] decreases by 0.6 units.
- The y-intercept of this line is at the origin [tex]\((0, 0)\)[/tex].
3. Determine the Region:
The inequality [tex]\( y < -\frac{3}{5} x \)[/tex] means we are interested in all the points that lie below the line [tex]\( y = -\frac{3}{5} x \)[/tex]. Therefore, any point [tex]\((x, y)\)[/tex] satisfying this inequality will be located in the region below this line.
4. Example Calculation:
Let's take a specific value of [tex]\( x \)[/tex] to see what [tex]\( y \)[/tex]-values satisfy the inequality.
- Suppose [tex]\( x = 1 \)[/tex]:
[tex]\[
y < -\frac{3}{5} \times 1
\][/tex]
Calculating the right side, we get:
[tex]\[
y < -0.6
\][/tex]
Therefore, when [tex]\( x = 1 \)[/tex], any [tex]\( y \)[/tex] value less than [tex]\(-0.6\)[/tex] will satisfy the inequality.
In summary, the inequality [tex]\( y < -\frac{3}{5} x \)[/tex] represents the region below the line [tex]\( y = -\frac{3}{5} x \)[/tex] in the coordinate plane. For [tex]\( x = 1 \)[/tex], the value of [tex]\( y \)[/tex] must be less than [tex]\(-0.6\)[/tex] to satisfy the inequality.
