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Solving Linear Equations and Inequalities: Summary of Key Concepts

Module by: Wade Ellis, Denny Burzynski. E-mail the authors

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Summary: This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. In this chapter, the emphasis is on the mechanics of equation solving, which clearly explains how to isolate a variable. The goal is to help the student feel more comfortable with solving applied problems. Ample opportunity is provided for the student to practice translating words to symbols, which is an important part of the "Five-Step Method" of solving applied problems (discussed in modules ((Reference)) and ((Reference))). This module provides a summary of the key concepts of the chapter "Solving Linear Equations and Inequalities".

Note: Your browser may not currently support MathML. See our browser support page for additional details. You can always view the correct math in the PDF version.

Summary of Key Concepts

Identity ((Reference))

An equation that is true for all acceptable values of the variable is called identity. x+3=x+3 x+3=x+3 is an identity.

Contradiction ((Reference))

Contradictions are equations that are never true regardless of the value substituted for the variable. x+1=x x+1=x is a contradiction.

Conditional Equation ((Reference))

An equation whose truth is conditional upon the value selected for the variable is called a conditional equation.

Solutions and Solving an Equation ((Reference))

The collection of values that make an equation true are called the solutions of the equation. An equation is said to be solved when all its solutions have been found.

Equivalent Equations ((Reference), (Reference))

Equations that have precisely the same collection of solutions are called equivalent equations.
An equivalent equation can be obtained from a particular equation by applying the same binary operation to both sides of the equation, that is,

  1. adding or subtracting the same number to or from both sides of that particular equation.
  2. multiplying or dividing both sides of that particular equation by the same non-zero number.

Literal Equation ((Reference))

A literal equation is an equation that is composed of more than one variable.

Recognizing an Identity ((Reference))

If, when solving an equation, all the variables are eliminated and a true statement results, the equation is an identity.

Recognizing a Contradiction ((Reference))

If, when solving an equation, all the variables are eliminated and a false statement results, the equation is a contradiction.

Translating from Verbal to Mathematical Expressions ((Reference))

When solving word problems it is absolutely necessary to know how certain words translate into mathematical symbols.

Five-Step Method for Solving Word Problems ((Reference))

  1. Let x x (or some other letter) represent the unknown quantity.
  2. Translate the words to mathematics and form an equation. A diagram may be helpful.
  3. Solve the equation.
  4. Check the solution by substituting the result into the original statement of the problem.
  5. Write a conclusion.

Linear Inequality ((Reference))

A linear inequality is a mathematical statement that one linear expression is greater than or less than another linear expression.

Inequality Notation ((Reference))

> Strictly greater than < Strictly less than ≥ Greater than or equal to ≤ Less than equal to > Strictly greater than < Strictly less than ≥ Greater than or equal to ≤ Less than equal to

Compound Inequality ((Reference))

An inequality of the form

 a<x<b a<x<b

is called a compound inequality.

Solution to an Equation in Two Variables and Ordered Pairs ((Reference))

A pair of values that when substituted into an equation in two variables produces a true statement is called a solution to the equation in two variables. These values are commonly written as an ordered pair. The expression (a, b) (a, b) is an ordered pair. In an ordered pair, the independent variable is written first and the dependent variable is written second.

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