The problem involves finding the value of $x$ using the angle measures provided:

• $m \angle LMP = 3x$
• $m \angle LMN = 4x + 11$
• $m \angle NMP = 2x - 6$

The diagram shows that these angles are arranged around a point $M$, which suggests that they sum up to $180^\circ$. Thus, the equation becomes:

$3x + (4x + 11) + (2x - 6) = 180^\circ$

Now, let's solve for $x$:

1. Combine like terms: $3x + 4x + 2x + 11 - 6 = 180$ $9x + 5 = 180$

2. Subtract 5 from both sides: $9x = 175$

3. Divide both sides by 9: $x = \frac{175}{9} \approx 19.44$

So, the value of $x$ is approximately $19.44$.

Would you like more details or have any questions?

Related Questions:

1. How do you verify if the calculated value of $x$ satisfies the angles?
2. What is the measure of each angle when $x \approx 19.44$?
3. How would the solution change if the angles were around a complete circle (360°)?
4. How do we know when angles around a point sum up to 180° versus 360°?
5. What are supplementary and complementary angles?

Tip:

Always double-check if angles are forming a straight line or a complete circle when adding them up. This helps to determine if they sum up to $180^\circ$ or $360^\circ$.