Chapter 1: Equations and Inequalities – Linear and Rational Equations

Equations and Inequalities

Introduction

This chapter introduces the foundational concepts of equations and inequalities, focusing on linear equations, equations involving fractions, rational equations, and the classification of equations. These concepts are essential for solving mathematical problems in College Algebra.

Linear Equations

Definition of a Linear Equation

A linear equation in one variable is an equation that can be written in the form:

$ax + b = 0$

• a and b are real numbers.

• a ≠ 0 (otherwise, the equation is not linear in x).

Creating Equivalent Equations

To solve equations, we often transform them into equivalent forms using the following operations:

• Simplify expressions by removing grouping symbols and combining like terms.

• Add or subtract the same real number or variable expression on both sides of the equation.

• Multiply or divide both sides by the same nonzero quantity.

• Interchange the two sides of the equation.

Solving Linear Equations: Step-by-Step

1. Simplify the algebraic expression on each side by removing grouping symbols and combining like terms.

2. Collect all variable terms on one side and all constant terms on the other side.

3. Isolate the variable and solve.

4. Check the proposed solution in the original equation.

Example: Solving a Linear Equation

Solve and check the equation:

$4x + 5 = 29$

Solution:

$4x + 5 - 5 = 29 - 5$ $4x = 24$ $\frac{4x}{4} = \frac{24}{4}$ $x = 6$

Check:

$4(6) + 5 = 24 + 5 = 29$

Linear Equations Involving Fractions

Solving Linear Equations with Fractions

When equations contain fractions, it is often helpful to clear denominators by multiplying both sides by the least common denominator (LCD).

Example: Solving a Linear Equation with Fractions

Solve and check:

$\frac{x-3}{4} = \frac{5}{14} - \frac{x+5}{7}$

Step 1: Find the LCD (here, 28) and multiply both sides:

$28 \left( \frac{x-3}{4} \right) = 28 \left( \frac{5}{14} \right) - 28 \left( \frac{x+5}{7} \right)$ $7(x-3) = 2 \times 5 - 4(x+5)$ $7x - 21 = 10 - 4x - 20$ $7x - 21 = -4x - 10$ $7x + 4x = -10 + 21$ $11x = 11$ $x = 1$

Check:

$\frac{1-3}{4} = \frac{5}{14} - \frac{1+5}{7}$ $\frac{-2}{4} = \frac{5}{14} - \frac{6}{7}$ $-\frac{1}{2} = \frac{5}{14} - \frac{12}{14}$ $-\frac{1}{2} = -\frac{7}{14}$ $-\frac{1}{2} = -\frac{1}{2}$

Rational Equations

Definition and Solving Rational Equations

A rational equation is an equation that contains at least one variable in the denominator. Solving rational equations often involves finding a common denominator and checking for extraneous solutions (values that make any denominator zero).

Example: Solving a Rational Equation

Solve:

$\frac{5}{2x} = \frac{17}{18} - \frac{1}{3x}$

Step 1: Identify restrictions by setting denominators equal to zero:

• $2x = 0 \implies x = 0$

• $3x = 0 \implies x = 0$

Step 2: Multiply both sides by the LCD (here, 18x):

$18x \left( \frac{5}{2x} \right) = 18x \left( \frac{17}{18} \right) - 18x \left( \frac{1}{3x} \right)$ $9(5) = x(17) - 6$ $45 = 17x - 6$ $45 + 6 = 17x$ $51 = 17x$ $x = 3$

Check that $x = 3$ does not make any denominator zero.

Types of Equations: Identity, Conditional, and Inconsistent

Classification of Equations

• Conditional Equation: True for at least one real number (e.g., $4x + 5 = 29$).

• Identity: True for all real numbers (e.g., $2(x+1) = 2x + 2$).

• Inconsistent Equation: Not true for any real number (e.g., $x + 1 = x + 2$).

Example: Inconsistent Equation

Determine if $4x - 7 = 4(x - 1) + 3$ is an identity, conditional, or inconsistent equation.

$4x - 7 = 4x - 4 + 3$ $4x - 7 = 4x - 1$ $-7 = -1$

This is a false statement, so the equation is inconsistent.

Example: Identity

Determine if $7x + 9 = 9(x + 1) - 2x$ is an identity, conditional, or inconsistent equation.

$7x + 9 = 9x + 9 - 2x$ $7x + 9 = 7x + 9$

This is always true, so the equation is an identity.

Solving Applied Problems Using Mathematical Models

Application Example

Mathematical models can be used to solve real-world problems. For example, suppose a formula relates the intensity of a negative life event (x) to the average level of depression (D):

$D = 10x + 53$

If the average level of depression is 10, find the intensity of the event:

$10 = 10x + 53$ $10x = 10 - 53$ $10x = -43$ $x = -4.3$

Note: The actual numbers may vary based on the specific model provided in the problem.

Summary Table: Types of Equations

Type	Description	Example
Identity	True for all real numbers	$2(x+1) = 2x + 2$
Conditional	True for at least one real number	$4x + 5 = 29$
Inconsistent	Not true for any real number	$x + 1 = x + 2$

Additional info: Some steps and context were inferred to ensure completeness and clarity for self-study.