Identify solutions to inequalities

Learning Objectives Define what a solution to an inequality is. Test a given numerical value to determine if it is a solution to a linear inequality. Test a given numerical value to determine if it is a solution to a simple quadratic inequality. Differentiate between strict inequalities (<, >) and non-strict inequalities (≤, ≥) when testing boundary points. Explain the difference between a single solution and a solution set. Verify solutions involving integers, fractions, and negative numbers. Ever tried to fit your luggage into an airline's overhead bin? You're checking if its size is less than or equal to the maximum allowed—that's identifying a solution to an inequality! ✈️ This tutorial will teach you the fundamental skill of testing numbers to see...

TermDefinitionExample InequalityA mathematical statement that compares two expressions using an inequality symbol. It shows that the two sides are not necessarily equal.3x - 5 > 10 is an inequality. Other symbols include < (less than), ≥ (greater than or equal to), and ≤ (less than or equal to). SolutionA value for a variable that makes an inequality a true statement.For the inequality x + 2 > 5, the number x = 4 is a solution because 4 + 2 > 5 (which simplifies to 6 > 5) is a true statement. Solution SetThe complete collection of all possible solutions to an inequality. This is often an infinite range of numbers.For x > 3, the solution set includes 3.1, 4, 5, 100, and all other numbers greater than 3. Boundary PointThe value that separates the solutions from the non-sol...

The Substitution Principle Given an inequality f(x) > c and a test value x = a, substitute 'a' into the inequality to get f(a) > c. This is the core process for checking any potential solution. Replace the variable with the number you are testing. The Truth Test If f(a) > c is a true statement, then x = a is a solution. If it is a false statement, x = a is not a solution. After substituting and simplifying, you must evaluate whether the resulting statement is mathematically correct. This determines if the tested value is a solution. Boundary Point Inclusion Rule For inequalities with > or <, the boundary point is NOT a solution. For inequalities with \geq or \leq, the boundary point IS a solution. This rule helps you quickly check the specifi...