2. Equations and Inequalities

Solving Equations Involving Absolute Value

Other Types of Equations

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

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

Using U-Substitution On An Equation with Rational Exponents

Solve each equation for the specified variable. (Assume all denominators are nonzero.) d=k√h, for h

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

Solving a Radical Equation with One Cube Root

 Using U-Substitution On A Radical Equation

Solve each equation. See Example 7. (x^2+24)^1/4 = 3

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

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

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.

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 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|

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

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

Solve each equation. See Example 7.(x-3)^2/5=(4x)^1/5

Use the method described in Exercises 83–86, if applicable, and properties of
absolute value to solve each equation or inequality. (Hint: Exercises 99 and 100 can
be solved by inspection.) | 4x^2 - 23x - 6 | = 0

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

Solve each equation. See Examples 4–6. √4x-x+3=0

How to find the roots of a polynomials by factoring

Solving Radical Equations word problems

Absolute Value: Double Absolute Value Equations

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

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

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 each equation. See Examples 4–6. √(3√(2x+3)) = √(5x-6)

Solve absolute value equations

Solve each equation. See Examples 4–6. √(x+2)=1-√(3x+7)

Factoring by Grouping

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

Solve each equation. See Examples 4–6. ∜(3x+1)=1

Solving Trigonometric Equations Using the Quadratic Formula - Example 1

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

Solve each equation. See Examples 4–6. √(2x+5)-√(x+2)=1

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

Solving Radicals and Rational Exponents involving Real World Problems

Use the method described in Exercises 83–86, if applicable, and properties of
absolute value to solve each equation or inequality. (Hint: Exercises 99 and 100 can
be solved by inspection.) | 3x^2 - 14x | = 5

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

√(x + 8) - √(x - 4) = 2

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

Application Involving a Square Root Word Problem

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

Solve each equation. See Examples 4–6. ∛2x=∛(5x+2)

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

|x| = 8

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|

In Exercises 107–110, use graphs to find each set. (-2,1] ∩ [-1,3)

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

√(2x + 19) - 8 = x

Solve each equation. See Example 7. x^2/3 = 2x^1/3

Solve each equation. See Examples 4–6. x - √(2x+3) = 0

Solve each equation. See Examples 4–6. √3x=√(5x+1)-1

 Solving Equations with Two Radicals / Example 3.7

Solve each equation. ⁵√ 2x = ⁵√ 3x+2

Solve each equation. See Examples 4–6. √2x=√(3x+12)-2

Finding all the Zeros of a Polynomial - Example 2

Solving an Equation Involving Rational Exponents - Example 2

Solve each equation. See Examples 4–6. √(x+7)+3=√(x-4)

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

Solving Trigonometric Equations Using the Quadratic Formula - Example 3

Solve each equation or inequality. |7/2-3x| -9=0

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

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

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

√(x + 3) = x - 3

Solve each equation. See Examples 8 and 9. 2x^4-7x^2+5=0

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

Solve each equation. See Examples 4–6. √(2√(7x+2)) = √(3x+2)

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

Solve a Square Root Equation 

Solve each equation. See Examples 4–6. √x+2=√(4+7√x)

❖ 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

Solve each equation. See Examples 4–6. √x-√(x-5)=1

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

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)

In Exercises 85–90, find the x-intercepts of the graph of each equation. Then use the x-intercepts to match the equation with its graph. [The graphs are labeled (a) through (f).] 

y = 2(x + 2)^2 + 5(x + 2) - 3

Solve each equation. See Examples 4–6. ∜(x-15)=2

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

9x^4 = 25x^2 - 16

Use the method described in Exercises 83–86, if applicable, and properties of
absolute value to solve each equation or inequality. (Hint: Exercises 99 and 100 can
be solved by inspection.) | x^2 + 5x + 5 | = 1

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

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

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

|x + 1| + 5 = 3

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

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

Solve each equation or inequality. |5x-1| = |2x+3|

Solve each equation. See Examples 8 and 9. (x-1)^2/3+(x-1)^1/3 -12 = 0

Solve each equation. See Example 7. (3x+7)^1/3-(4x+2)^1/3=0

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

(y - 8/y)^2 + 5(y - 8/y) - 14 = 0

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

In Exercises 104–106, express each interval in set-builder notation and graph the interval on a number line. (-2, ∞)

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

Solve each equation. See Examples 4–6. ∜(x^2+2x)= ∜3

Solve each equation. See Examples 4–6. √(2x-5)=2+√(x-2)

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

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

Solve each equation. See Example 7. x^3/2 = 125

Write an equation involving absolute value that says the distance 
between p and q is 2 units.

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. See Example 7. (x-3)^2/5 = 4

Use the method described in Exercises 83–86, if applicable, and properties of
absolute value to solve each equation or inequality. (Hint: Exercises 99 and 100 can
be solved by inspection.) | x2 + 1 | - | 2x | = 0

Solve each equation. See Example 7. 3x^3/4 = x^1/2

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

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

Solve each equation. See Examples 4–6. √(3x+7) = 3x+5

Solve each equation. See Examples 4–6. √(4x+13) = 2x-1

Solve each equation or inequality. |x+10| = |x-11|

In Exercises 107–110, use graphs to find each set. [1,3) ∩ (0,4)

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

|2x - 1| = 5

Match each equation in Column I with the correct first step for solving it in Column II. √(x+5) = 7

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

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

Use the method described in Exercises 83–86, if applicable, and properties of
absolute value to solve each equation or inequality. (Hint: Exercises 99 and 100 can
be solved by inspection.) | x^2 - 9 | = x + 3

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

|x - 2| = 7

Solve each equation. See Examples 4–6. √(4x+1)-√(x-1)=2

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

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

Solve each equation. See Examples 8 and 9. x^-2/3+x^-1/3-6=0

Match each equation in Column I with the correct first step for solving it in Column II. (x+5)^2/3 - (x+5)^1/3 - 6 = 0

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)

In Exercises 85–90, find the x-intercepts of the graph of each equation. Then use the x-intercepts to match the equation with its graph. [The graphs are labeled (a) through (f).] 

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

In Exercises 85–90, find the x-intercepts of the graph of each equation. Then use the x-intercepts to match the equation with its graph. [The graphs are labeled (a) through (f).] 

y = √(x - 4) + √(x + 4) - 4

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 equation for the specified variable. (Assume all denominators are nonzero.) x^2/3+y^2/3=a^2/3, for y

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

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

Factor Polynomial Given a Complex / Imaginary Root

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

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

Solve each equation. |3 - 2x | = |5 - 2x |

Solve each equation. See Example 7. x^5/4 = 32

Solving Absolute Value Equations - Example 1

Solving an Absolute Value Equation with Fractions

Solve each equation. See Examples 4–6. 3-√x=√(2√(x)-3)

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

Solve each equation. See Examples 8 and 9. 4(x+1)^4-13(x+1)^2=-9

Solving Equations Involving Square Roots

Solve each equation or inequality. |x+4| = 7

Solving an Equation Containing Two Radicals - Example 1

Solve each equation. See Examples 4–6. √(6x+7) - 9 = x-7

In Exercises 101–106, solve each equation.

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

Solve each equation. See Examples 4–6. ∛(4x+3)=∛(2x-1)

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).

Learn the toughest concepts covered in Algebra with step-by-step video tutorials and practice problems by world-class tutors