Which statement about the following system of inequalities is true?

[tex]\[
\left\{
\begin{array}{l}
y - 2x \leq 2.5 \\
y + x^2 + 0.1x \leq -0.6
\end{array}
\right.
\][/tex]

A. There is no solution because the shading does not overlap.
B. There is no solution because the graphs do not intersect.
C. The solution contains points in four quadrants of the coordinate plane.
D. The solution is equal to the solution to [tex]\( y + x^2 + 0.1x \leq -0.6 \)[/tex].



Answer :

To determine which statement about the given system of inequalities is true, we need to analyze their graphs and potential points of intersection.

The system of inequalities is:
[tex]\[ \left\{\begin{array}{l} y - 2x \leq 2.5 \\ y + x^2 + 0.1x \leq -0.6 \end{array}\right. \][/tex]

1. Convert the inequalities to equations to find potential intersection points:

[tex]\[ y - 2x = 2.5 \quad \text{(Equation 1)} \][/tex]
[tex]\[ y + x^2 + 0.1x = -0.6 \quad \text{(Equation 2)} \][/tex]

2. Isolate [tex]\( y \)[/tex] in both equations:

[tex]\[ y = 2x + 2.5 \quad \text{(From Equation 1)} \][/tex]
[tex]\[ y = -x^2 - 0.1x - 0.6 \quad \text{(From Equation 2)} \][/tex]

3. Set the two expressions for [tex]\( y \)[/tex] equal to each other to find [tex]\( x \)[/tex]:

[tex]\[ 2x + 2.5 = -x^2 - 0.1x - 0.6 \][/tex]

4. Rearrange the equation into standard quadratic form:

[tex]\[ x^2 + 2.1x + 3.1 = 0 \][/tex]

5. Calculate the determinant ([tex]\( D \)[/tex]) of the quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = 2.1 \)[/tex], and [tex]\( c = 3.1 \)[/tex]:

The determinant is given by the formula:
[tex]\[ D = b^2 - 4ac \][/tex]

6. Substitute the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] into the determinant formula:

[tex]\[ D = (2.1)^2 - 4(1)(3.1) \][/tex]
[tex]\[ D = 4.41 - 12.4 \][/tex]
[tex]\[ D = -7.99 \][/tex]

7. Analyze the determinant:

If the determinant [tex]\( D \)[/tex] is negative (i.e., [tex]\( D < 0 \)[/tex]), it means the quadratic equation has no real solutions. This implies the equations do not intersect at any real points.

Since the determinant [tex]\( D \)[/tex] is [tex]\(-7.99\)[/tex] and is negative, the graphs of the two equations indeed do not intersect.

Given this analysis, we can conclude:

There is no solution because the graphs do not intersect.