2.1.1 Solving Linear Equations and Inequalities
Key Objectives • Write and solve linear equations using a variety of methods. • Solve linear equations with variables on both sides. • Identify identities and contradictions. • Solve and graph linear inequalities. Key Terms • An equation is a mathematical statement that two expressions are equal.
• The solution set of an equation is the value or values of the variable that make the equation true. • A linear equation in one variable is an equation that can be written in the form ax = b , where a and b are constants and a ≠ 0. • An inequality is a mathematical statement that shows the relationship between quantities that are not equivalent using one of the following symbols: < , > , ≤ , ≥ , or ≠ . • An identity is an equation or inequality that is true for all values of the variable. The equation or inequality is always true. • A contradiction is an equation or inequality that has no solutions. The equation or inequality is never true. In a linear equation, the variable is not under a radical sign and is not raised to a power other than 1. The variable is also not an exponent and is not in a denominator. The following equations are examples of linear equations in one variable. 4 x = 8 2 x − 5 = 0.1 x + 2 Example 1 Travel Application Solving a linear equation requires isolating the variable on one side of the equation. To isolate the variable, perform the inverse, or opposite, of every operation in the equation on both sides of the equation. Apply the inverse operations in the reverse order of the order of operations.
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