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Consider the system of inequalities and its graph:

[tex]\[
\begin{array}{l}
y \leq -0.75x \\
y \leq 3x - 2
\end{array}
\][/tex]

In which section of the graph does the actual solution to the system lie?

1
2
3
4

Sagot :

GinnyAnswer
To determine the section of the graph where the actual solution to the system of inequalities lies, we first need to understand the given inequalities:

1. [tex]\( y \leq -0.75x \)[/tex]
2. [tex]\( y \leq 3x - 2 \)[/tex]

Next, let's find the intersection point of the lines:
- The first inequality corresponds to the line [tex]\( y = -0.75x \)[/tex].
- The second inequality corresponds to the line [tex]\( y = 3x - 2 \)[/tex].

To find the intersection point, set the two equations equal to each other:
[tex]\[ -0.75x = 3x - 2 \][/tex]

Solving for [tex]\( x \)[/tex] involves the following steps:
1. Add [tex]\( 0.75x \)[/tex] to both sides:
[tex]\[ 0 = 3.75x - 2 \][/tex]
2. Add 2 to both sides:
[tex]\[ 2 = 3.75x \][/tex]
3. Divide both sides by 3.75:
[tex]\[ x = \frac{2}{3.75} \][/tex]
[tex]\[ x = 0.533 \][/tex]

Now, substitute [tex]\( x = 0.533 \)[/tex] back into one of the original equations to find [tex]\( y \)[/tex]. Let's use the equation [tex]\( y = -0.75x \)[/tex]:
[tex]\[ y = -0.75 \times 0.533 \][/tex]
[tex]\[ y = -0.4 \][/tex]

So, the intersection point is approximately [tex]\( (0.533, -0.4) \)[/tex].

Now, we interpret the given inequalities:
- [tex]\( y \leq -0.75x \)[/tex] implies the region below the line [tex]\( y = -0.75x \)[/tex].
- [tex]\( y \leq 3x - 2 \)[/tex] implies the region below the line [tex]\( y = 3x - 2 \)[/tex].

The solution to the system of inequalities will be the region where these two regions overlap.

To determine which section of the graph this overlapping region lies in, recognize that:
- The line [tex]\( y = -0.75x \)[/tex] has a negative slope and starts from the origin, angling downwards to the right.
- The line [tex]\( y = 3x - 2 \)[/tex] has a positive slope and intersects the y-axis at [tex]\( y = -2 \)[/tex].

The intersection point [tex]\( (0.533, -0.4) \)[/tex] lies in the Cartesian (x, y) coordinate plane.

Considering the geometrical arrangement of the lines and their slopes, the overlapping region (where both inequalities are satisfied) will be the region that lies below both lines. This region is typically in the lower-right section of the Cartesian plane.

Therefore, the actual solution to the system of inequalities lies in section [tex]\( \boxed{4} \)[/tex].
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