1. Equations & Inequalities

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)

Solve each equation for x. a²x + 3x =2a²

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)

Find all values of x satisfying the given conditions. y₁ = 5(2x - 8) - 2, y₂ = 5(x - 3) + 3, and y₁ = y₂.

In Exercises 61–66, find all values of x satisfying the given conditions. $y_1 = \frac{x - 3}{5}, \quad y_2 = \frac{x - 5}{4}, \quad \text{and} \quad y_1 - y_2 = 1$

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.

In Exercises 67–70, find all values of x such that y = 0.

y = 2[3x - (4x - 6)] - 5(x - 6)

In Exercises 67–70, find all values of x such that y = 0. $y = \frac{x + 6}{3x - 12} - \frac{5}{x - 4} - \frac{2}{3}$

Find all values of x such that y = 0. y = 1/(5x + 5) - 3/(x + 1) + 7/5

In Exercises 71–78, solve each equation. Then determine whether the equation is an identity, a conditional equation, or an inconsistent equation. 4(x + 5) = 21 + 4x

Exercises 73–75 will help you prepare for the material covered in the next section. Simplify: √18 - √8

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. 4/(x - 2) + 3/(x + 5) = 7/(x + 5)(x - 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. 4x/(x + 3) - 12/(x - 3) = (4x² + 36)/(x² - 9)