2. Equations and Inequalities

Solving Equations Involving Absolute Value

Other Types of Equations

Absolute Value: Double Absolute Value Equations

 Solving a Radical Equation with a Square Root Set Equal to a Variable

Using U-Substitution On An Equation with Rational Exponents

Solving a Radical Equation with One Cube Root

 Using U-Substitution On A Radical Equation

Solve each radical equation in Exercises 88–89. √ (2x-3) + x = 3

Solve each equation in Exercises 96–102 by the method of your choice. 2√(x-1) = x

If 5 times a number is decreased by 4, the principal square root of this difference is 2 less than the number. Find the number(s).

In Exercises 91–100, find all values of x satisfying the given conditions.

y = |2 - 3x| and y = 13

Solve each equation in Exercises 41–60 by making an appropriate substitution.

x^(2/5) + x^(1/5) - 6 = 0

In Exercises 137–140, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.

The equation |x| = - 6 is equivalent to x = 6 or x = - 6.

Solve each equation in Exercises 96–102 by the method of your choice. x^3 + 2x^2 = 9x + 18

Solve each equation in Exercises 92–93 by making an appropriate substitution. x^4 - 5x^2 + 4 = 0

Solve each equation in Exercises 41–60 by making an appropriate substitution.

x - 13√x + 40 = 0

Solving an Equation Involving Rational Exponents - Example 1

Solving Algebraic Equations With Roots and Exponents

The rule for rewriting an absolute value equation without absolute value bars can be extended to equations with two sets of absolute value bars:
If u and v represent algebraic expressions, then |u| = |v| is equivalent to u = v or u = - v. Use this to solve the equations in Exercises 77–84.

|4x - 3| = |4x - 5|

Solve: √(6x - 2) = √(2x + 3) - √(4x - 1).

Exercises 177–179 will help you prepare for the material covered in the next section.

If - 8 is substituted for x in the equation 5x^(2/3) + 11x^(1/3) + 2 = 0, is the resulting statement true or false?

Finding all the Zeros of a Polynomial - Example 1

How to find the roots of a polynomials by factoring

Solving Radical Equations word problems

If a number is decreased by 3, the principal square root of this difference is 5 less than the number. Find the number(s).

Solving Equations with Rational Exponents

Solving Trigonometric Equations Using the Quadratic Formula - Example 2

Solve each polynomial equation in Exercises 1–10 by factoring and then using the zero-product principle.

4y^3 - 2 = y - 8y^2 

Solve each radical equation in Exercises 11–30. Check all proposed solutions.

√(6x + 1) = x - 1

Solve each radical equation in Exercises 11–30. Check all proposed solutions.

√(1 + 4√x) = 1 + √x

Solving Equations Involving Rational Exponents

In Exercises 91–100, find all values of x satisfying the given conditions.

y = x^3 + 4x^2 - x + 6 and y = 10

Solving an Equation Involving a Single Radical (Square Root) - Example 2

 Solving Equations with Two Radicals / Example 3.8

Solve absolute value equations

Factoring by Grouping

Solving Absolute Value Equations Containing TWO Absolute Value Expressions - Example 1

Solving Trigonometric Equations Using the Quadratic Formula - Example 1

Solving Word Problems in Distance, Rate, and Time Using Quadratics - Example 1

Solving Absolute Value Equations Containing TWO Absolute Value Expressions - Example 2

Solving Radicals and Rational Exponents involving Real World Problems

Solve each radical equation in Exercises 11–30. Check all proposed solutions.

√(x + 8) - √(x - 4) = 2

Application Involving a Square Root Word Problem

The rule for rewriting an absolute value equation without absolute value bars can be extended to equations with two sets of absolute value bars:
If u and v represent algebraic expressions, then |u| = |v| is equivalent to u = v or u = - v. Use this to solve the equations in Exercises 77–84.

|3x - 1| = |x + 5|

Solve each radical equation in Exercises 11–30. Check all proposed solutions.

√(2x + 19) - 8 = x

 Solving Equations with Two Radicals / Example 3.7

Finding all the Zeros of a Polynomial - Example 2

Solving an Equation Involving Rational Exponents - Example 2

Determining the Correct U-Substitution (If Possible) / Example 4.4a

Solving Trigonometric Equations Using the Quadratic Formula - Example 3

Solving Absolute Value Equations Containing TWO Absolute Value Expressions - Example 3

Solve each radical equation in Exercises 11–30. Check all proposed solutions.

√(x + 3) = x - 3

Solve for x: x^(5/6) + x^(2/3) - 2x^(1/2) = 0

 Determining the Correct U-Substitution (If Possible) / Example 4.4b

Solve a Square Root Equation 

❖ Finding all the Zeros of a Polynomial - Example 3 ❖

Solve each polynomial equation in Exercises 1–10 by factoring and then using the zero-product principle.

5x^4 - 20x^2 = 0

Solve each polynomial equation in Exercises 86–87. 2x^4 = 50 x^2

Find extraneous solutions in algebraic radical equations

 Solving a Radical Equation with One Square Root

Solve each equation in Exercises 41–60 by making an appropriate substitution.

(x - 5)^2 - 4(x - 5) - 21 = 0

In Exercises 61–76, solve each absolute value equation or indicate that the equation has no solution.

|2x - 1| + 3 = 3

A Projectile Problem Using Quadratics - Example 1

Solve radical algebraic equations

Using U-Substitution Instead of Multiplying Out

Solve each equation in Exercises 41–60 by making an appropriate substitution.

9x^4 = 25x^2 - 16

Determining the Correct U-Substitution (If Possible) / Example 4.4c

The rule for rewriting an absolute value equation without absolute value bars can be extended to equations with two sets of absolute value bars:
If u and v represent algebraic expressions, then |u| = |v| is equivalent to u = v or u = - v. Use this to solve the equations in Exercises 77–84.

|x^2 - 6| = |5x|

Solve each polynomial equation in Exercises 1–10 by factoring and then using the zero-product principle.

3x^4 = 81x

In Exercises 61–76, solve each absolute value equation or indicate that the equation has no solution.

|x + 1| + 5 = 3

Linear Equation, Word Problem #3 : Interest Earned on Two Investments

Solve each equation in Exercises 96–102 by the method of your choice. -4|x+1| + 12 = 0

 Solving Equations with Rational Exponents

Solving an Equation Containing Two Radicals - Example 2

In Exercises 61–76, solve each absolute value equation or indicate that the equation has no solution.

2|4 - (5/2)x| + 6 = 18

How to Factor and Solve fourth Degree Polynomials - Simple Technique

Factoring Out the GCF - Positive Exponents

Solve each equation in Exercises 41–60 by making an appropriate substitution.

x^(-2) - x^(-1) - 6 = 0

Solving an Equation Involving a Single Radical (Square Root) - Example 3

In Exercises 101–106, solve each equation.

|x^2 + 2x - 36| = 12

In Exercises 91–100, find all values of x satisfying the given conditions.

y = (x - 5)^(3/2) and y = 125

Solve the equations containing absolute value in Exercises 94–95. |2x+1| = 7

In Exercises 61–76, solve each absolute value equation or indicate that the equation has no solution.

|2x - 1| = 5

Solve each polynomial equation in Exercises 1–10 by factoring and then using the zero-product principle.

4x^3 - 12x^2 = 9x - 27

Solving an Equation Involving a Single Radical (Square Root) - Example 1

In Exercises 61–76, solve each absolute value equation or indicate that the equation has no solution.

|x - 2| = 7

Solve each equation with rational exponents in Exercises 31–40. Check all proposed solutions.

(x - 4)^(2/3) = 16

The rule for rewriting an absolute value equation without absolute value bars can be extended to equations with two sets of absolute value bars:
If u and v represent algebraic expressions, then |u| = |v| is equivalent to u = v or u = - v. Use this to solve the equations in Exercises 77–84.

|2x^2 - 4| = |2x^2|

In Exercises 101–106, solve each equation.

|√x - 5| = 2

Solve Equations with Rational Exponents (Two Solutions)

Solve each equation with rational exponents in Exercises 31–40. Check all proposed solutions.

(x^2 - x - 4)^(3/4) - 2 = 6

In Exercises 61–76, solve each absolute value equation or indicate that the equation has no solution.

2|3x - 2| = 14

Solve each radical equation in Exercises 11–30. Check all proposed solutions.

√(2x + 3) + √(x - 2) = 2

In Exercises 61–76, solve each absolute value equation or indicate that the equation has no solution.

7|5x| + 2 = 16

Factor Polynomial Given a Complex / Imaginary Root

In Exercises 91–100, find all values of x satisfying the given conditions.

y = x - √(x - 2) and y = 4

Solving Absolute Value Equations - Example 2

Solving Absolute Value Equations - Example 1

Solving an Absolute Value Equation with Fractions

In Exercises 91–100, find all values of x satisfying the given conditions.

y1 = 6(2x/(x - 3))^2, y2 = 5(2x/(x - 3)), and y1 exceeds y2 by 6

Linear Equation, Word Problem #6 : Concentration of a Mixture

Solving Equations Involving Square Roots

Solving an Equation Containing Two Radicals - Example 1

In Exercises 101–106, solve each equation.

x(x + 1)^3 - 42(x + 1)^2 = 0

Solving an Equation Containing Two Radicals - Example 3

Solve each equation with rational exponents in Exercises 31–40. Check all proposed solutions.

(x - 4)^(3/2) = 27

Solve without squaring both sides:

5 - (2/x) = √(5 - 2/x).

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