Question

solve for x.
1)
2)

Explanation:

Step1: Use the exterior angle theorem

The exterior angle theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non - adjacent interior angles.
For problem 1), we have \(11x + 2=(7x - 4)+48\).

Step2: Simplify the equation

First, simplify the right - hand side of the equation: \((7x - 4)+48=7x+44\). So the equation becomes \(11x + 2=7x+44\).
Subtract \(7x\) from both sides: \(11x-7x + 2=7x-7x+44\), which gives \(4x+2 = 44\).
Subtract 2 from both sides: \(4x+2-2=44 - 2\), so \(4x=42\).
Divide both sides by 4: \(x=\frac{42}{4}=\frac{21}{2}=10.5\).

For problem 2), use the triangle angle - sum theorem. The sum of the interior angles of a triangle is \(180^{\circ}\). Let the third angle be \(A\), then \(A + 47^{\circ}+72^{\circ}=180^{\circ}\), so \(A=180-(47 + 72)=61^{\circ}\).
Since the triangle has an angle \(A = 61^{\circ}\) and the side opposite to it is \(8x-3\), we assume it is an isosceles or other type (but using the angle - sum first). If we assume it is a general triangle and we want to solve for \(x\) in terms of the side (assuming it is a side - angle relationship, but if we consider the problem might be a mis - labeled exterior - angle - like problem (if we assume the side \(8x - 3\) is related to the angles). But if we use the angle - sum:
The third angle \(=180-(47 + 72)=61^{\circ}\). If we assume \(8x-3\) is related to the angle (maybe a mis - draw, if we assume it is a side and we use the Law of Sines (but no other side information). If we assume it is a problem where \(8x-3\) is the measure of the third angle (if mis - labeled): \(8x-3=61\).
Add 3 to both sides: \(8x-3 + 3=61+3\), so \(8x=64\).
Divide both sides by 8: \(x = 8\).

Answer:

1. \(x = 10.5\)
2. \(x = 8\)