To find the [tex]\( x \)[/tex]-coordinate of the point that divides the directed line segment [tex]\(\overline{AB}\)[/tex] in a given ratio, we used the formula:
[tex]\[ x = \left(\frac{a}{a+b}\right)(x_2 - x_1) + x_1 \][/tex]
Here are the specific values we are working with:
- The coordinates of [tex]\( A \)[/tex] are [tex]\( (-4, -10) \)[/tex].
- The coordinates of [tex]\( B \)[/tex] are [tex]\( (-11, -7) \)[/tex].
- The ratio [tex]\( a:b \)[/tex] is given as [tex]\( 3:4 \)[/tex].
Substituting these values into our formula, we get:
[tex]\[
x = \left( \frac{3}{3+4} \right) (-11 - (-4)) + (-4)
\][/tex]
Let's simplify this step-by-step:
1. Compute the denominator of the fraction: [tex]\( 3 + 4 = 7 \)[/tex].
2. Compute the difference in the [tex]\( x \)[/tex]-coordinates: [tex]\( -11 - (-4) = -11 + 4 = -7 \)[/tex].
3. Multiply the ratio fraction by this difference: [tex]\( \frac{3}{7} \times (-7) = -3 \)[/tex].
4. Add this result to the [tex]\( x \)[/tex]-coordinate of point [tex]\( A \)[/tex]: [tex]\( -3 + (-4) = -7 \)[/tex].
Thus, the [tex]\( x \)[/tex]-coordinate of the point that divides [tex]\(\overline{AB}\)[/tex] into a [tex]\( 3:4 \)[/tex] ratio is:
[tex]\[
\boxed{-7}
\][/tex]
