Question

check understanding

1. the equation ( x+(75.3 - x)=75.3 ) represents the sum of two angle measures.

how many possible combinations of angle measures satisfy these conditions?

1. two bicyclists on a 75 - mile trail ride toward each other. one bicyclist

begins at the 45 - mile marker. the other begins at the end of the trail. the
expressions shown represent the distance from the parking lot for each
bicyclist when they meet.
a. write an equation to represent the bicyclists meeting after ( x ) hours.

b. after how many hours do the bicyclists meet?

c. how far away are the bicyclists from the
parking lot when they meet?

Explanation:

1.

Step1: Simplify the equation

$$\begin{align*} x+(75.3 - x)&=75.3\\ x + 75.3-x&=75.3\\ 75.3&=75.3 \end{align*}$$

This is an identity. For any real - valued \(x\), the equation holds. Since \(x\) represents an angle measure (\(0

Step1: Set up the equation

When the two bicyclists meet, the sum of the distances they have traveled is equal to the length of the segment between their starting points. The first bicyclist starts at the \(45\) - mile marker and travels at a rate (assuming a rate of \(10\) miles per hour from the expressions \(45 + 10x\) and \(75-10x\)) and the second starts at the \(75\) - mile marker. So, \(45 + 10x=75-10x\)

Step1: Solve the equation \(45 + 10x=75-10x\)

Add \(10x\) to both sides:
\(45+10x + 10x=75-10x+10x\)
\(45 + 20x=75\)
Subtract \(45\) from both sides:
\(20x=75 - 45\)
\(20x=30\)
Divide both sides by \(20\):
\(x=\frac{30}{20}=\frac{3}{2}=1.5\)

Answer:

Infinitely many.

2.

A.