To determine the value of [tex]\( x \)[/tex] for which the lengths of the sides of a triangle [tex]\( 2x \)[/tex], [tex]\( x + 4 \)[/tex], and 13 inches add up to a total perimeter of 35 inches, let's follow these steps:
1. Understand the problem statement:
- The lengths of the sides of the triangle are given as [tex]\( 2x \)[/tex], [tex]\( x + 4 \)[/tex], and 13 inches.
- The perimeter of the triangle is the sum of the lengths of its three sides.
- The perimeter is given as 35 inches.
2. Set up the equation for the perimeter:
- Perimeter of a triangle = Sum of the lengths of its sides.
[tex]\[
\text{Perimeter} = 2x + (x + 4) + 13
\][/tex]
- According to the given problem, the perimeter is 35 inches.
[tex]\[
2x + (x + 4) + 13 = 35
\][/tex]
3. Simplify the equation:
- Combine like terms on the left-hand side.
[tex]\[
2x + x + 4 + 13 = 35
\][/tex]
[tex]\[
3x + 17 = 35
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
- To isolate [tex]\( x \)[/tex], first subtract 17 from both sides of the equation.
[tex]\[
3x + 17 - 17 = 35 - 17
\][/tex]
[tex]\[
3x = 18
\][/tex]
- Divide both sides by 3.
[tex]\[
x = \frac{18}{3}
\][/tex]
[tex]\[
x = 6
\][/tex]
Therefore, the value of [tex]\( x \)[/tex] is [tex]\( \boxed{6} \)[/tex].
5. Verify the solution:
- Calculate the lengths of the sides using [tex]\( x = 6 \)[/tex]:
[tex]\[
2x = 2(6) = 12
\][/tex]
[tex]\[
x + 4 = 6 + 4 = 10
\][/tex]
[tex]\[
13 \, (\text{given})
\][/tex]
- Check the perimeter:
[tex]\[
12 + 10 + 13 = 35
\][/tex]
- The calculated perimeter matches the given perimeter, confirming that [tex]\( x = 6 \)[/tex] is correct.
