The image provided contains a problem setup with the following information:

• $DE = 4x - 16$
• $EF = 8$
• $DF = \frac{7x + 19}{2}$

The geometry setup is given by the equation:

$DE + EF = DF$

To solve for $x$, we can substitute the given expressions into the equation.

Step 1: Substitute the values into the equation

$(4x - 16) + 8 = \frac{7x + 19}{2}$

Step 2: Simplify the left-hand side

$4x - 16 + 8 = \frac{7x + 19}{2}$

$4x - 8 = \frac{7x + 19}{2}$

Step 3: Eliminate the fraction by multiplying both sides by 2

$2(4x - 8) = 7x + 19$

$8x - 16 = 7x + 19$

Step 4: Solve for $x$

Subtract $7x$ from both sides:

$x - 16 = 19$

Add 16 to both sides:

$x = 35$

Step 5: Find $DE$ and $DF$

• For $DE$: $DE = 4x - 16 = 4(35) - 16 = 140 - 16 = 124$

• For $DF$: $DF = \frac{7x + 19}{2} = \frac{7(35) + 19}{2} = \frac{245 + 19}{2} = \frac{264}{2} = 132$

So, $x = 35$, $DE = 124$, and $DF = 132$, which matches the values in the image.

Do you have any further questions or want additional details?

Related Questions:

1. How would the equation change if $DF$ were different?
2. How do you derive the equation for segments in a line geometry problem?
3. What happens if $EF$ were unknown? How would you solve for it?
4. Can you solve for $x$ using another method like graphing?
5. How does this method generalize to more complex geometric setups?

Tip: Always check your algebraic simplifications to avoid errors when working with expressions like fractions.