Let's proceed step-by-step to find the length of the segment [tex]\(\overline{JK}\)[/tex].
1. Understand the given information:
- [tex]\(JK = 2x - 1\)[/tex]
- [tex]\(IK = 3x + 2\)[/tex]
- [tex]\(IJ = 3x - 5\)[/tex]
We know from the given information that point [tex]\(J\)[/tex] is on the line segment [tex]\(\overline{IK}\)[/tex]. This implies that the lengths of the segments [tex]\(JK\)[/tex] and [tex]\(IJ\)[/tex] should sum up to the total length of [tex]\(IK\)[/tex]:
[tex]\[
JK + IJ = IK
\][/tex]
2. Set up the equation:
Write this relationship as an equation using the given expressions for the segments:
[tex]\[
(2x - 1) + (3x - 5) = 3x + 2
\][/tex]
3. Combine like terms:
Let's combine like terms on the left-hand side:
[tex]\[
2x - 1 + 3x - 5 = 3x + 2
\][/tex]
[tex]\[
5x - 6 = 3x + 2
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
Isolate [tex]\(x\)[/tex] by first subtracting [tex]\(3x\)[/tex] from both sides:
[tex]\[
5x - 3x - 6 = 2
\][/tex]
[tex]\[
2x - 6 = 2
\][/tex]
Add 6 to both sides to further isolate [tex]\(2x\)[/tex]:
[tex]\[
2x = 8
\][/tex]
Finally, divide by 2:
[tex]\[
x = 4
\][/tex]
5. Find the length of [tex]\(JK\)[/tex]:
Substitute [tex]\(x = 4\)[/tex] back into the expression for [tex]\(JK\)[/tex]:
[tex]\[
JK = 2x - 1
\][/tex]
[tex]\[
JK = 2(4) - 1 = 8 - 1 = 7
\][/tex]
Therefore, the length of [tex]\(\overline{JK}\)[/tex] is [tex]\(\boxed{7}\)[/tex].
