Answer :
To solve for [tex]\( D F \)[/tex] given that [tex]\( D \)[/tex] is between [tex]\( E \)[/tex] and [tex]\( F \)[/tex], and given the lengths [tex]\( E D = 6x - 4 \)[/tex], [tex]\( D F = 3x + 5 \)[/tex], and [tex]\( E F = 46 \)[/tex], we follow these steps:
1. Set up the equation:
Since [tex]\( D \)[/tex] is between [tex]\( E \)[/tex] and [tex]\( F \)[/tex], we can write the following equation:
[tex]\[
E D + D F = E F
\][/tex]
Substituting the given expressions, we get:
[tex]\[
(6x - 4) + (3x + 5) = 46
\][/tex]
2. Combine like terms:
Combine the terms involving [tex]\( x \)[/tex] and the constants:
[tex]\[
6x - 4 + 3x + 5 = 46
\][/tex]
Simplify this:
[tex]\[
9x + 1 = 46
\][/tex]
3. Solve for [tex]\( x \)[/tex]:
Subtract 1 from both sides to isolate the term with [tex]\( x \)[/tex]:
[tex]\[
9x = 45
\][/tex]
Divide both sides by 9:
[tex]\[
x = 5
\][/tex]
4. Find [tex]\( D F \)[/tex]:
Now that we have [tex]\( x \)[/tex], we can substitute it back into the equation for [tex]\( D F \)[/tex]:
[tex]\[
D F = 3x + 5
\][/tex]
Substituting [tex]\( x = 5 \)[/tex]:
[tex]\[
D F = 3(5) + 5 = 15 + 5 = 20
\][/tex]
Therefore, the length of [tex]\( D F \)[/tex] is [tex]\( \boxed{20} \)[/tex].
1. Set up the equation:
Since [tex]\( D \)[/tex] is between [tex]\( E \)[/tex] and [tex]\( F \)[/tex], we can write the following equation:
[tex]\[
E D + D F = E F
\][/tex]
Substituting the given expressions, we get:
[tex]\[
(6x - 4) + (3x + 5) = 46
\][/tex]
2. Combine like terms:
Combine the terms involving [tex]\( x \)[/tex] and the constants:
[tex]\[
6x - 4 + 3x + 5 = 46
\][/tex]
Simplify this:
[tex]\[
9x + 1 = 46
\][/tex]
3. Solve for [tex]\( x \)[/tex]:
Subtract 1 from both sides to isolate the term with [tex]\( x \)[/tex]:
[tex]\[
9x = 45
\][/tex]
Divide both sides by 9:
[tex]\[
x = 5
\][/tex]
4. Find [tex]\( D F \)[/tex]:
Now that we have [tex]\( x \)[/tex], we can substitute it back into the equation for [tex]\( D F \)[/tex]:
[tex]\[
D F = 3x + 5
\][/tex]
Substituting [tex]\( x = 5 \)[/tex]:
[tex]\[
D F = 3(5) + 5 = 15 + 5 = 20
\][/tex]
Therefore, the length of [tex]\( D F \)[/tex] is [tex]\( \boxed{20} \)[/tex].