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McGraw Hill Glencoe Algebra 2, 2012

MH

McGraw Hill Glencoe Algebra 2, 2012 View details

6. Solving Rational Equations and Inequalities

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Exercise 10 Page 576

a Alicia traveled

20

miles with the wind and

20

miles against it. The formula that relates distance to time is

d = r t,

or

t = \frac{d}{r}

. Therefore, we can write the expression for Alicia's time with the wind as follows.

t = \frac{d}{r} \Leftrightarrow \frac{2 0}{1 1 . 5 + x}

b Because the

r a t e

will be slower when Alicia is biking against the wind, the expression for her time can be written as follows.

t = \frac{d}{r} \Leftrightarrow \frac{2 0}{1 1 . 5 - x}

c Alicia's total trip will be her biking with the wind and against it. Therefore, to find the total time we will sum up the time it takes her to bike with and against the wind.

Time with the wind + Time against the wind = Total time ⇕ 1 hr 20 min + 2 hr 30 min = 3 hr 50 min

It will take Alicia

3

hours and

50

minutes, or

3 \frac{5}{6}

hours to complete the trip.

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}

Multiply

\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}

Multiply

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.

Subchapter links

Check Your Understanding p.576

H.O.T. Problems p.577

Standardized Test Practice p.578

Spiral Review p.578

Skills Review p.578