d The speed of the wind either increases or decreases the rate at which Alicia bikes. Therefore, the variable

x

in Part A and Part B represents the speed of the wind.

Time with the wind + Time against the wind = Total time ⇕ \frac{2 0}{1 1 . 5 + x} + \frac{2 0}{1 1 . 5 - x} = 3 \frac{5}{6}

Let's solve this rational equation for

x

to determine the speed of the wind.

\frac{2 0}{1 1 . 5 + x} + \frac{2 0}{1 1 . 5 - x} = 3 \frac{5}{6}

Write fraction as a mixed number

\frac{2 0}{1 1 . 5 + x} + \frac{2 0}{1 1 . 5 - x} = \frac{2 3}{6}

▼

Simplify

Multiply $\frac{2 0}{1 1 . 5 + x}$ by $\frac{1 1 . 5 - x}{1 1 . 5 - x}$

\frac{2 0}{1 1 . 5 + x} (\frac{1 1 . 5 - x}{1 1 . 5 - x}) + \frac{2 0}{1 1 . 5 - x} = \frac{2 3}{6}

Multiply $\frac{2 0}{1 1 . 5 - x}$ by $\frac{1 1 . 5 + x}{1 1 . 5 + x}$

\frac{2 0}{1 1 . 5 + x} (\frac{1 1 . 5 - x}{1 1 . 5 - x}) + \frac{2 0}{1 1 . 5 - x} (\frac{1 1 . 5 + x}{1 1 . 5 + x}) = \frac{2 3}{6}

\frac{2 0 ( 1 1 . 5 - x )}{( 1 1 . 5 + x ) ( 1 1 . 5 - x )} + \frac{2 0 ( 1 1 . 5 + x )}{( 1 1 . 5 - x ) ( 1 1 . 5 + x )} = \frac{2 3}{6}

Add terms

\frac{2 0 ( 1 1 . 5 - x ) + 2 0 ( 1 1 . 5 + x )}{( 1 1 . 5 + x ) ( 1 1 . 5 - x )} = \frac{2 3}{6}

Distribute $20$

\frac{2 3 0 - 2 0 x + 2 3 0 + 2 0 x}{( 1 1 . 5 + x ) ( 1 1 . 5 - x )} = \frac{2 3}{6}

Distribute $(11.5 + x)$

\frac{2 3 0 - 2 0 x + 2 3 0 + 2 0 x}{1 1 . 5 ^{2} - 1 1 . 5 x + 1 1 . 5 x - x ^{2}} = \frac{2 3}{6}

Calculate power

\frac{2 3 0 - 2 0 x + 2 3 0 + 2 0 x}{4 6 0 - 1 1 . 5 x + 1 1 . 5 x - x ^{2}} = \frac{2 3}{6}

Add and subtract terms

\frac{4 6 0}{1 3 2 . 2 5 - x ^{2}} = \frac{2 3}{6}

Cross multiply

460 (6) = 23 (132.25 - x^{2})

Distribute $23$

460 (6) = 23 (132.25) - 23 x^{2}

2760 = 3041.75 - 23 x^{2}

$LHS - 3041.75 = RHS - 3041.75$

- 281.75 = - 23 x^{2}

$LHS / - 23 = RHS / - 23$

12.25 = x^{2}

Calculate root

3.5 = x

The speed of the wind is

3.5

mph.

Exercise