2. Equations and Inequalities

Solve Rational Equations with Variables in the Denominators

Linear Equations and Rational Equations

Solving a Rational Equation - Ex 4

What is an identity equation? Give an example.

Solve each equation for x. 2(x-a) +b =3x+a

In Exercises 1–26, solve and check each linear equation.

3(x - 1) = 21

In Exercises 15–35, solve each equation. Then state whether the equation is an identity, a conditional equation, or an inconsistent equation. 3-5(2x + 1) - 2(x-4) = 0

In Exercises 15–35, solve each equation. Then state whether the equation is an identity, a conditional equation, or an inconsistent equation. (3x+1)/3 - 13/2 = (1-x)/4

In Exercises 1–26, solve and check each linear equation.

3(x - 8) = x

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.

5/2x - 8/9 = 1/18 - 1/3x

Solving Linear Equations - Example 1

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

❖ Solving Linear Equations Made Easy! ❖

An Intro to Solving Linear Equations: Solving some Basic Linear Equations

Solve each equation. 6(3x-1)= 8 - (10x-14)

Exercises 27–40 contain linear equations with constants in denominators. Solve each equation.

3x/5 - (x - 3)/2 = (x + 2)/3

Word Problem: Finding Consecutive Numbers That Satisfy a Given Requirement - Ex 2

Understand a linear equation with infinite solutions

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

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)

Solve each equation. 0.08x+0.06(x+12) = 7.72

In Exercises 61–66, find all values of x satisfying the given conditions.

y1 = (2x - 1)/(x^2 + 2x - 8), y2 = 2/(x + 4), y3 = 1/(x - 2), and y1 + y2 = y3.

Solving a Basic Rational Equation - Ex 2

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.

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)

In Exercises 1–26, solve and check each linear equation.

4x + 9 = 33

Solve and check: 24 + 3 (x + 2) = 5(x − 12).

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)

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 equations with no solutions

Solve each equation. 4[2x-(3-x)+5] = -6x - 28

Solving Problems using Mathematical Modeling Techniques Part 1 (MathAngel369)

Determine whether each equation is an identity, a conditional equation, or a contradic-tion. Give the solution set. 1/2(6x+20) = x+4 +2(x+3)

Solving Multi - Step Linear Equations

Solve each formula for the specified variable. Assume that the denominator is not 0 if variables appear in the denominator. F = GMm/r², for m (force of gravity)

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

Use inverse operations to solve equations of form ax+b=c

In Exercises 1–26, solve and check each linear equation.

4(x + 9) = x

Solving a Basic Linear Equation - Example 3

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 + 3x - 10) - 1/(x^2 + x - 6) = 3/(x^2 - x - 12)

Exercises 73–75 will help you prepare for the material covered in the next section.

Rationalize the denominator: (7 + 4√2)/(2 - 5√2).

Solving Two - Step Linear Equations

Solve each problem. If x represents the number of pennies in a jar in an applied problem, which of the following equations cannot be a correct equation for finding x? (Hint:Solve the equations and consider the solutions.)
A. 5x+3 =11
B.12x+6 =-4
C.100x =50(x+3)
D. 6(x+4) =x+24

In Exercises 61–66, find all values of x satisfying the given conditions.

y1 = 5(2x - 8) - 2, y2 = 5(x - 3) + 3, and y1 = y2.

Solving a Basic Rational Equation - Ex 5

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

In Exercises 15–35, solve each equation. Then state whether the equation is an identity, a conditional equation, or an inconsistent equation. 2(x-4)+3(x+5)=2x-2

Finding extraneous solutions in rational algebraic equations

Finding Original Price of an Item Knowing the Price after a Discount

In Exercises 15–35, solve each equation. Then state whether the equation is an identity, a conditional equation, or an inconsistent equation. 7x + 13 = 2(2x-5) + 3x + 23

Solve rational equations using additive and multiplicative inverses

Linear Equation, Word Problem #2 : Consecutive Even Integers

Find b such that (7x + 4)/b + 13 = x has a solution set given by {- 6}.

Solving Linear Equations

In Exercises 1–26, solve and check each linear equation.

2 - (7x + 5) = 13 - 3x

In Exercises 1–26, solve and check each linear equation.

2x - 7 = 6 + x

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 linear equations with rational numbers by using integers

Decide whether each statement is true or false. The solution set of 2x+5=x -3 is {-8}.

Solve multi-step equations: using the distributive property

Solving a Basic Rational Equation - Ex 1

Solve linear equations with rational numbers using mixed numbers

In Exercises 99–106, solve each equation.

- 2{7 - [4 -2(1 - x) + 3]} = 10 - [4x - 2(x - 3)]

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 99–106, solve each equation.

5 - 12x = 8 - 7x - [6 ÷ 3(2 + 5^3) + 5x]

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.

4/x = 5/2x + 3

Definition of Linear Equation In One Variable

In Exercises 1–26, solve and check each linear equation.

11x - (6x - 5) = 40

In Exercises 1–26, solve and check each linear equation.

7x - 5 = 72

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

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

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

What is a conditional equation? Give an example.

In Exercises 67–70, 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.

5x + 7 = 2x + 7

Solving Linear Equations with Fractional Quantities

Decide whether each statement is true or false. The equation 5x=4x is an example of a contradiction.

Solve Linear Equations by Factoring and Using the LCD

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

y = (x + 6)/(3x - 12) - 5/(x - 4) - 2/3

In Exercises 15–35, solve each equation. Then state whether the equation is an identity, a conditional equation, or an inconsistent equation. 2x-5 = 7

Solving a Basic Linear Equation - Example 2

Linear Equation, Word Problem #5: Distance, Rate, Time Example

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.

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

In Exercises 99–106, solve each equation.

[(3 + 6)^2 ÷ 3] × 4 = - 54 x

In Exercises 15–35, solve each equation. Then state whether the equation is an identity, a conditional equation, or an inconsistent equation. 7(x-4) = x + 2

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 + 4) - 7 = - 4/(x + 4)

Solve rational equations by eliminating denominators

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) = (4x^2 + 36)/(x^2 - 9)

Exercises 27–40 contain linear equations with constants in denominators. Solve each equation.

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

Linear Equations – Example (Ages)

Solving a Basic Rational Equation - Ex 3

Solving Linear Equations Requiring the Multiplication of Binomials

In Exercises 71–78, solve each equation. Then determine whether the equation is an identity, a conditional equation, or an inconsistent equation.

4x + 7 = 7(x + 1) - 3x

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

Linear Equation, Word Problem #4 : Side Lengths of a Triangle

In Exercises 1–26, solve and check each linear equation.

16 = 3(x - 1) - (x - 7)

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

Evaluate x^2 - x for the value of x satisfying 4(x - 2) + 2 = 4x - 2(2 - x).

In Exercises 99–106, solve each equation.

4x + 13 - {2x - [4(x - 3) - 5]} = 2(x - 6)

An Intro to Solving Linear Equations:  What Does it Mean to be a Solution?

Solving Rational Equations

Solve each equation. A= 24f / B(p+1), for f (approximate annual interest rate)

Exercises 27–40 contain linear equations with constants in denominators. Solve each equation.

20 - x/3 = x/2

Compare relationships between quantities using linear models

Solving Two - Step Linear Equations, Another Example

Solve linear equations with rational numbers using fractions

Solve each formula for the specified variable. Assume that the denominator is not 0 if variables appear in the denominator. s = 1/2gt², for g (distance traveled by a falling object)

Solve each equation. 1/15(2x+5) = 1/9(x+2)

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 27–40 contain linear equations with constants in denominators. Solve each equation.

(x + 3)/6 = 3/8 + (x - 5)/4

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

In Exercises 1–26, solve and check each linear equation.

7x + 4 = x + 16

Exercises 27–40 contain linear equations with constants in denominators. Solve each equation.

x/5 - 1/2 = x/6

Determine whether each equation is an identity, a conditional equation, or a contradic-tion. Give the solution set. 2(x-8) = 3x-16

In Exercises 1–26, solve and check each linear equation.

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

2/(x - 2) = x/(x - 2) - 2

Exercises 27–40 contain linear equations with constants in denominators. Solve each equation.

x/3 = x/2 - 2

Exercises 73–75 will help you prepare for the material covered in the next section.

Simplify: √18 - √8

Exercises 27–40 contain linear equations with constants in denominators. Solve each equation.

3x/5 = 2x/3 + 1

In Exercises 61–66, find all values of x satisfying the given conditions.

y1 = (x - 3)/5, y2 = (x - 5)/4, and y1 - y2 = 1.

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

In Exercises 1–26, solve and check each linear equation.

x - 5(x + 3) = 13

Solve fraction problems by writing and solving equations

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)

Retaining the Concepts. Solve and determine whether 8(x - 3) + 4 = 8x - 21 is an identity, a conditional equation, or an inconsistent equation.

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

What is an inconsistent equation? Give an example.

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

Predict how many solutions a linear equation has

Rates and Time Required To Complete a Task

In Exercises 71–78, solve each equation. Then determine whether the equation is an identity, a conditional equation, or an inconsistent equation.

10x + 3 = 8x + 3

Linear Equation , Word Problem #8 :  Work, Rates, Time To Complete a Task

Determine whether each equation is an identity, a conditional equation, or a contradic-tion. Give the solution set. -0.6(x-5)+0.8(x-6) = 0.2x - 1.8

❖ Solving a Linear System of Equations by Graphing ❖

Linear Equation, Word Problem #1 : Averages

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