Solving Linear Equations and Inequalities

• Learning Objectives
• Verify linear solutions
• Use equality properties to solve basic linear equations
• Clear fractions from equations
• Identify linear inequalities and check solutions
• Solve linear inequalities and express solutions graphically on a number line using interval notation.

Solving Basic Linear Equations

• An equation indicates that two algebraic expressions are equal.
• A linear equation with one variable, x, can be in the form ax + b = 0, where a and b are real numbers and a ≠ 0.

Solution

• A solution to a linear equation is any value that replaces the variable to produce a true statement.
• To verify a solution, substitute the value for x and check if a true statement is obtained.
• Alternatively, when an equation equals a constant, you can verify a solution by substituting the value for the variable and showing the result equals that constant.
• Solutions "satisfy the equation".

Equivalent Equations

• Equivalent equations have the same solution set. Properties of equality are used to obtain equivalent equations where A and B are algebraic expressions and c is a nonzero number.

Properties of Equality and Solving Equations

• Addition Property: If A = B, then A + c = B + c
• Subtraction Property: If A = B, then A - c = B - c.
• Multiplication Property: If A = B, then Ac = Bc
• Division Property: If A = B, then A/c = B/c

Isolating Variables

• Avoid multiplying or dividing both sides of an equation by 0, as division by 0 is undefined and multiplying by 0 results in 0 = 0.
• Algebraic equations are solved by isolating the variable with a coefficient of 1.
• For a linear equation in the form ax + b = c, use the equality property of addition or subtraction to isolate the variable term first
• Then isolate the actual variable via multiplication or division.
• Symmetric is where 4 = y is equivalent to y = 4

General Guidelines for Solving Linear Equations

• Simplify both sides of the equation by order of operations and combining like terms.
• Use equality properties to combine like terms on opposite sides, to get the variable term on one side and the constant term on the other.
• Add or subtract to isolate the variable
• Divide or multiply to isolate the variable
• Check whether the answer solves the original equation.
• Combine same-side like terms and simplify before solving.

Linear Equations with Multiple Terms

• Combine like terms on opposite sides of the equal sign, using addition or subtraction to place like terms on the same side.
• For equations with fractional coefficients, multiply both sides by the least common multiple of the denominators to clear the fractions and obtain integer coefficients. This technique works only for equations - do not clear fractions when simplifying expressions.

Applications Involving Linear Equations

• Algebra simplifies solving real-world problems using letters to represent unknowns, restating problems as equations, and systematic solution techniques.
• Translate the problem's wording into mathematical statements that describe relationships, by identifying key words and phrases while carefully reading the problem.
• Let x represent the unknown and state it in words.

Important Key Words

• Sum: increased by, more than, plus, added to, total
• Difference: decreased by, subtracted from, less, minus
• Product: multiplied by, of, times, twice
• Quotient: divided by, ratio, per
• "Is": total, result

Steps for Word Problems

• Read and identify key words and phrases, and also organize given information.
• Assign a letter or expression to each unknown variable.
• Translate and set up an algebraic equation that models the problem.
• Solve the resulting equation
• Answer the question in sentence form, making sure it makes sense and can be checked.

Linear Inequalities

• A linear inequality relates a linear expression as either less than or greater than another.
• A solution to a linear inequality is a real number that produces a true statement when substituted for the variable.
• Linear inequalities have infinitely many or no solutions
• Solutions are graphed on a number line and/or use interval notation.
• Expressing Solutions: The number line is shaded to incorporate the solutions.
• An open parenthesis indicates the number itself is non inclusive, ie x>3
• The left bracket symbol, [, would show that the endpoint is included. ie x≤1 can have one as a solution

Interval Notation

• Express x>3 as (3,∞). The infinity symbol ∞ is not an actual number.
• Express x≤1 in interval notation as (-∞,1]. Negative infinity indicated with −∞.
• Notation for both the number line and the interval notation requires that ∞ and −∞ always use parentheses, never brackets

Steps for Linear Inequalities

• Most techniques for solving inequalities are similar to equations, although there are some differences
• Add or subtract any real number to both sides of an inequality
• Multiply or divide both sides by any positive real number to create equivalent inequalities.
• It is helpful to confirm whether solutions are accurate by testing a few values in and out of the solution

Negative numbers with Inequalities

• An inequality must be reversed when multiplying or dividing by a negative number
• Equivalent inequalities have the same solution set: the variable is isolated and the process is similar to solving linear equations.

Addition Property Of Inequalities

• If A< B, then A + c < B + c

Subtraction Property Of Inequalities

• If A< B, then A − c < B − c

Multiplication Property Of Inequalities:

• If c is positive and A< B, then Ac < Bc.
• If c is negative and A< B, then Ac > Bc

Division Property Of Inequalities:

• If c is positive and A< B, then A/c< B/c
• If c is negative and A< B, then A/c > B/c

A number is "at least" 5 is expressed a ≥ 5.