1. Equations & Inequalities

Find all values of x satisfying the given conditions. y₁ = (2x - 1)/(x² + 2x - 8), y₂ = 2/(x + 4), y₃ = 1/(x - 2), and y₁ + y₂ = y₃.

Distance to the Horizon The 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?

In Exercises 61–66, find all values of x satisfying the given conditions. y₁ = 5/(x + 4), y₂ = 3/(x + 3), y₃ = (12x + 19)/(x² + 7x + 12). and y₁ + y₂ = y₃.

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. 1/(x - 1) + 5 = 11/(x - 1)

Solve and check each linear equation. 4x + 9 = 33

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. 3/(2x - 2) + 1/2 = 2/(x - 1)

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. 3/(x + 2) + 2/(x - 2) = 8/(x + 2)(x - 2)

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

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

In Exercises 99–106, solve each equation. 0.7x + 0.4(20) = 0.5(x + 20)

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

Write each formula as an English phrase using the word varies or proportional. V = 1/3 πr²h, where V is the volume of a cone of radius r and height h

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

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.

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