Linear Equation and Inequality

LINEAR EQUATION IN ONE VARIABLE X

• is an equation that can be written in the form ax + b = 0, where a and b are real numbers, and a ≠ 0.
• Solving an equation in x involves determining all values of x that result in a true statement when substituted into the equation. Such values are solutions, or roots, of the equation
• Solution set - the set of all such solutions
• Equivalent equations - two or more equations that have the same solution set

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What are the differences between what I’m supposed to do with algebraic expressions and algebraic equations?

• We simplify algebraic expressions. We solve algebraic equations.

Solving a Linear equation

1. Simplify the algebraic expression on each side by removing grouping symbols and combining like terms.
2. Collect all the variable terms on one side and all the numbers, or constant terms, on the other side.
3. Isolate the variable and solve.
4. Check the proposed solution in the original equation.

Rational Equation

• an equation containing one or more rational expressions.

Least Common Denominator

• a polynomial consisting of the product of all prime factors in the denominators, with each factor raised to the greatest power of its occurrence in any denominator.

Empty Set (∅)

• When there is no solution to the equation
• The solution set for a certain equation contains no elements

TYPES OF EQUATIONS

1. Identity - an equation that is true for all real numbers for which both sides are defined
2. Conditional Equation - an equation that is not an identity, but that is true for at least one real number
3. Inconsistent Equation - an equation that is not true for even one real number

Solving an inequality

• is the process of finding the set of numbers that make the inequality a true statement.
• These numbers are called the solutions of inequality and we say that they satisfy the inequality.
• The set of all solutions is called the solution set of the inequality.
• Set-builder notation and a new notation, called interval notation, are used to represent these solution sets.

Interval Notation

• Some sets of real numbers can be represented using interval notation
• The open interval (a, b) represents the set of real numbers between, but not including, a and b.
• The closed interval [a, b] represents the set of real numbers between, and including, a and b.
• The infinite interval (a, ∞) represents the set of real numbers that are greater than a.
• The infinite interval ( - ∞ , b] represents the set of real numbers that are less than or equal to b.

Parentheses and Brackets in Interval Notation

• Parentheses indicate endpoints that are not included in an interval. Square brackets indicate endpoints that are included in an interval. Parentheses are always used with ∞ or - ∞ .

Linear Inequality

• A linear inequality in x can be written in one of the following forms: ax + b < 0, ax + b ≤ 0, ax + b > 0, ax + b ≥  0. In each form, a ≠ 0.

• Inequalities with the same solution set are said to be equivalent

• If you attempt to solve an inequality that has no solution, you will eliminate the variable and obtain a false statement, such as 0 > 1. If you attempt to solve an inequality that is true for all real numbers, you will eliminate the variable and obtain a true statement, such as 0 < 1.

• Compound Inequality - is an inequality that combines two simple inequalities (combined by the word and*)*