1. Equations & Inequalities

In Exercises 1–26, solve and check each linear equation. 3(x - 1) = 21

In Exercises 1–34, solve each rational equation. If an equation has no solution, so state. (7x−4)/5x = 9/5 − 4/x

In Exercises 1–14, simplify the expression or solve the equation, whichever is appropriate. 3(2x-5)-2(4x+1)=-5(x+3)-2

In Exercises 1–26, solve and check each linear equation. x - 5(x + 3) = 13

Solve each equation. 5/6x - 2x+ 4/3=5/3

In Exercises 1–26, solve and check each linear equation. 3(x - 8) = x

In Exercises 1–34, solve each rational equation. If an equation has no solution, so state. 8/x²−9 + 4/x+3 = 2/x−3

In Exercises 1–26, solve and check each linear equation. 2(x - 1) + 3 = x - 3(x + 1)

Solve each equation. -4(2x-6) +8x= 5x+24+x

In Exercises 1–26, solve and check each linear equation. 25 - [2 + 5y - 3(y + 2)] = - 3(2y - 5) - [5(y - 1) - 3y + 3]

In Exercises 15–35, solve each equation. Then state whether the equation is an identity, a conditional equation, or an inconsistent equation. 2x/3 = 6 - x/4

Exercises 27–40 contain linear equations with constants in denominators. Solve each equation. 20 - x/3 = x/2

In Exercises 25-38, solve each equation. 3x/5 = 2x/3 +1

In Exercises 1–34, solve each rational equation. If an equation has no solution, so state. 4/(x²+3x−10) + 1/(x²+9x+20) = 2/(x²+2x−8)

Exercises 27–40 contain linear equations with constants in denominators. Solve each equation. 3x/5 - x = x/10 - 5/2

Determine whether each equation is an identity, a conditional equation, or a contradic-tion. Give the solution set. -6(2x+1) - 3(x-4) = -15x+1

Solve each formula for the specified variable. Assume that the denominator is not 0 if variables appear in the denominator. I=Prt,for P (simple interest)

Exercises 41–60 contain rational equations with variables in denominators. For each equation, a. write the value or values of the variable that make a denominator zero. These are the restrictions on the variable. b. Keeping the restrictions in mind, solve the equation. 7/2x - 5/3x = 22/3

Solve each formula for the specified variable. Assume that the denominator is not 0 if variables appear in the denominator. z = x-μ/σ, for x (standardized value)

Exercises 41–60 contain rational equations with variables in denominators. For each equation, a. write the value or values of the variable that make a denominator zero. These are the restrictions on the variable. b. Keeping the restrictions in mind, solve the equation. 2/(x - 2) = x/(x - 2) - 2

Solve each equation for x. 3x=(2x-1)(m+4)

In Exercises 61–66, find all values of x satisfying the given conditions. y1 = 5/(x + 4), y2 = 3/(x + 3), y3 = (12x + 19)/(x^2 + 7x + 12). and y1 + y2 = y3.

In Exercises 71–78, solve each equation. Then determine whether the equation is an identity, a conditional equation, or an inconsistent equation. 5x + 9 = 9(x + 1) - 4x

Exercises 73–75 will help you prepare for the material covered in the next section. Rationalize the denominator: (7 + 4√2)/(2 - 5√2).

The equations in Exercises 79–90 combine the types of equations we have discussed in this section. Solve each equation. Then state whether the equation is an identity, a conditional equation, or an inconsistent equation. 2/x + 1/2 = 3/4

Evaluate x^2 - (xy - y) for x satisfying 3(x + 3)/5 = 2x + 6 and y satisfying - 2y - 10 = 5y + 18.

Solve: x/(x−3)=2x/(x−3)−5/3

In Exercises 99–106, solve each equation. - 2{7 - [4 -2(1 - x) + 3]} = 10 - [4x - 2(x - 3)]

Solve: 3x-1/5 + x+2/2 = -3/10. (Section 1.4, Example 4)

Find b such that (4x - b)/(x - 5) = 3 has a solution set given by {Ø}.

In Exercises 35–54, solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? 1/R = 1/R1 + 1/R2 for R1

Solve each equation. |2x + 3/ 3x - 4 | = 1

Solve each equation. | 5/ x-3 | = 10

Solve each equation. |3/ 2x - 1 | = 4

Solve each equation. | 6x + 1/ x - 1 | = 3

Determine the values of the variable that cannot possibly be solutions of each equation. Do not solve. See Examples 1 and 2. (5/(2x+3))-(1/(x-6))=0

Solve each equation. x/x+2 + 1/x+3 = 2/x²+2x

Solve each equation. 2/x+2 + 1/x+4 = 4/x²+6x+8

Determine the values of the variable that cannot possibly be solutions of each equation. Do not solve. See Examples 1 and 2. 3/(x-2) + 1/(x+1) = 3/(x^2-x-2)

Determine the values of the variable that cannot possibly be solutions of each equation. Do not solve. See Examples 1 and 2. 1/(4x) - 2/x = 3

Determine the values of the variable that cannot possibly be solutions of each equation. Do not solve. See Examples 1 and 2. 5/(2x) - 2/x = 6

Solve each equation. See Example 1. (2x+5)/2 - 3x/(x-2) = x

Solve each equation. See Example 1. (4x+3)/4 - 2x/(x+1) = x

Solve each equation. See Example 1. x/(x-3) = 3/(x-3) + 3

Solve each equation. See Example 1. x/(x-4) = 4/(x-4) + 4

Solve each equation. See Example 1. 3/(x^2+x-2) - 1/(x^2-1) = 7/(2x^2+6x+4)

Solve each equation. See Example 2. (4x+3)/(x+1) + 2/x = 1/(x^2+x)

Solve each equation. See Example 2. x/(x-1) - 1/(x+1) = 2/(x^2-1)

Solve each equation. See Example 2. (x+4)/2x = (x-1)/3

Solve each equation. See Example 2. 2x/(x-2) = 5 + 4x^2/(x-2)

Solve each equation. See Example 2. 3x^2/(x-1) + 2 = x/(x-1)

Solve each equation for the specified variable. (Assume all denominators are nonzero.) 1/R=1/r_1 + 1/r_2, for R

Solve each equation. 10/4x-4 = 1 /1-x

In Exercises 45–46, describe in words the variation shown by the given equation. z = kx^2 √y

In Exercises 45–46, describe in words the variation shown by the given equation. z = k√x / y^2

In Exercises 11–20, write an equation that expresses each relationship. Then solve the equation for y. x varies directly as z and inversely as the sum of y and w.

In Exercises 11–20, write an equation that expresses each relationship. Then solve the equation for y. x varies directly as z and inversely as the difference between y and w.

In Exercises 11–20, write an equation that expresses each relationship. Then solve the equation for y. x varies jointly as z and the difference between y and w.

In Exercises 11–20, write an equation that expresses each relationship. Then solve the equation for y. x varies jointly as z and the sum of y and w.

In Exercises 11–20, write an equation that expresses each relationship. Then solve the equation for y. x varies jointly as y and z and inversely as the square of w.

In Exercises 11–20, write an equation that expresses each relationship. Then solve the equation for y. x varies jointly as y and z and inversely as the square root of w.

In Exercises 11–20, write an equation that expresses each relationship. Then solve the equation for y. x varies directly as the cube root of z and inversely as y.

In Exercises 11–20, write an equation that expresses each relationship. Then solve the equation for y. x varies directly as the cube of z and inversely as y.

In Exercises 11–20, write an equation that expresses each relationship. Then solve the equation for y. x varies jointly as y and z.

Use the four-step procedure for solving variation problems given on page 447 to solve Exercises 1–10. y varies jointly as a and b and inversely as the square root of c. y = 12 when a = 3, b = 2, and c = 25. Find y when a = 5, b = 3 and c = 9.

Use the four-step procedure for solving variation problems given on page 447 to solve Exercises 1–10. y varies jointly as x and z. y = 25 when x = 2 and z = 5. Find y when x = 8 and z = 12.

Use the four-step procedure for solving variation problems given on page 447 to solve Exercises 1–10. y varies directly as x and inversely as the square of z. y = 20 when x = 50 and z = 5. Find y when x = 3 and z = 6.

Use the four-step procedure for solving variation problems given on page 447 to solve Exercises 1–10. y varies inversely as x. y = 12 when x = 5. Find y when x = 2.

Solve the variation problems in Exercises 77–82. The pitch of a musical tone varies inversely as its wavelength. A tone has a pitch of 660 vibrations per second and a wavelength of 1.6 feet. What is the pitch of a tone that has a wavelength of 2.4 feet?

Solve the variation problems in Exercises 77–82. The distance that a body falls from rest is directly proportional to the square of the time of the fall. If skydivers fall 144 feet in 3 seconds, how far will they fall in 10 seconds?

Write each formula as an English phrase using the word varies or proportional. C=2πr, where C is the circumference of a circle of radius r.

Write each formula as an English phrase using the word varies or proportional. r = d/t, where r is the speed when traveling d miles in t hours.

Write each formula as an English phrase using the word varies or proportional. V = 1/3 πr^2h, where V is the volume of a cone of radius r and height h

Solve each problem. Circumference of a CircleThe circumference of a circle varies directly as the radius. A circle with radius 7 in. has circumference 43.96 in. Find the circumference of the circle if the radius changes to 11 in.

Solve each problem. Resistance of a WireThe resistance in ohms of a platinum wire temperature sensor varies directly as the temperature in kelvins (K). If the resistance is 646 ohms at a temperature of 190 K, find the resistance at a temperature of 250 K.

Solve each problem. Distance to the HorizonThe distance that a person can see to the horizon on a clear day from a point above the surface of Earth varies directly as the square root of the height at that point. If a person 144 m above the surface of Earth can see 18 km to the horizon, how far can a person see to the horizon from a point 64 m above the surface?

Fill in the blank(s) to correctly complete each sentence, or answer the question as appropriate. In the equation y = 6x, y varies directly as x. When x=5, y=30. What is the value of y when x=10?

Using k as the constant of variation, write a variation equation for each situation. h varies inversely as t.

Solve each problem. If y varies directly as x, and y=20 when x=4, find y when x = -6.

Solve each problem. If m varies jointly as x and y, and m=10 when x=2 and y=14, find m when x=21 and y=8.

Solve each problem. If y varies inversely as x, and y=10 when x=3, find y when x=20.

Solve each problem. Suppose r varies directly as the square of m, and inversely as s. If r=12 when m=6 and s=4, find r when m=6 and s=20.

Solve each problem. Let a be directly proportional to m and n^2, and inversely proportional to y^3. If a=9when m=4, n=9, and y=3, find a when m=6, n=2, and y=5.

Match each statement with its corresponding graph in choices A–D. In each case, k > 0. y varies directly as x. (y=kx)

Match each statement with its corresponding graph in choices A–D. In each case, k > 0. y varies directly as the second power of x. (y=kx^2)

Solve each problem. Hooke's Law for a SpringHooke's law for an elastic spring states that the distance a spring stretches varies directly as the force applied. If a force of 15 lb stretches a certain spring 8 in., how much will a force of 30 lb stretch the spring?

Solve each problem. The speed of a pulley varies inversely as its diameter. One kind of pulley, with diameter 3 in., turns at 150 revolutions per minute. Find the speed of a similar pulley with diameter 5 in.

Solve each problem. Simple InterestSimple interest varies jointly as principal and time. If $1000 invested for 2 yr earned $70, find the amount of interest earned by $5000 invested for 5 yr.

Solve each problem. Current FlowIn electric current flow, it is found that the resistance offered by a fixed length of wire of a given material varies inversely as the square of the diameter of the wire. If a wire 0.01 in. in diameter has a resistance of 0.4 ohm, what is the resistance of a wire of the same length and material with diameter 0.03 in., to the nearest ten-thousandth of an ohm?

Solve each problem. Force of WindThe force of the wind blowing on a vertical surface varies jointly as the area of the surface and the square of the velocity. If a wind of 40 mph exerts a force of 50 lb on a surface of 1/2 ft^2, how much force will a wind of 80 mph place on a surface of 2 ft^2?

Solve each problem. Period of a PendulumThe period of a pendulum varies directly as the square rootof the length of the pendulum and inversely as the square root of the accelerationdue to gravity. Find the period when the length is 121 cm and the acceleration due to gravity is 980 cm per second squared, if the period is 6π seconds when the length is 289 cm and the acceleration due to gravity is 980 cm per second squared.

Solve each problem. Nuclear Bomb DetonationSuppose the effects of detonating a nuclear bomb will be felt over a distance from the point of detonation that is directly proportional to the cube root of the yield of the bomb. Suppose a 100-kiloton bomb has certain effects to a radius of 3 km from the point of detonation. Find, to the nearest tenth, the dis-tance over which the effects would be felt for a 1500-kiloton bomb.

Answer each question. What happens to y if y varies directly as x, and x is halved?