To determine the [tex]\( x \)[/tex]-coordinate of the point that divides the directed line segment from [tex]\( J \)[/tex] to [tex]\( K \)[/tex] in a ratio of 2.5, we follow these steps:
1. Identify the coordinates of points [tex]\( J \)[/tex] and [tex]\( K \)[/tex]. Let's set:
[tex]\[
J = (x_1) = -4
\][/tex]
[tex]\[
K = (x_2) = 4
\][/tex]
2. Let's denote the ratio as [tex]\( m:n \)[/tex] where [tex]\( m = 2.5 \)[/tex] and [tex]\( n = 1 \)[/tex].
3. Use the formula to find the x-coordinate of the point dividing the segment in the ratio [tex]\( m:n \)[/tex]:
[tex]\[
x = \left(\frac{m}{m+n}\right) \left(x_2 - x_1\right) + x_1
\][/tex]
4. Substitute the values into the formula:
[tex]\[
x = \left(\frac{2.5}{2.5 + 1}\right) \left(4 - (-4)\right) + (-4)
\][/tex]
5. Simplify the expression inside the parenthesis:
[tex]\[
x = \left(\frac{2.5}{3.5}\right) \left(4 + 4\right) - 4
\][/tex]
[tex]\[
x = \left(\frac{2.5}{3.5}\right) \times 8 - 4
\][/tex]
6. Calculate the fraction:
[tex]\[
\frac{2.5}{3.5} = \frac{5}{7}
\][/tex]
7. Continue with the simplified values:
[tex]\[
x = \left(\frac{5}{7}\right) \times 8 - 4
\][/tex]
8. Multiply the fraction by 8:
[tex]\[
x = \frac{5 \times 8}{7} - 4
\][/tex]
[tex]\[
x = \frac{40}{7} - 4
\][/tex]
9. Simplify the fraction and calculate the final result:
[tex]\[
x \approx 5.714285714285714 - 4
\][/tex]
[tex]\[
x \approx 1.7142857142857144
\][/tex]
Therefore, the [tex]\( x \)[/tex]-coordinate of the point that divides the line segment from [tex]\( J \)[/tex] to [tex]\( K \)[/tex] into the ratio of 2.5 is approximately [tex]\( 1.7142857142857144 \)[/tex].
