Final answer:
Proving the solution of an inequality is valid involves listing known information, solving the inequality, and checking the solution by substituting it back into the original inequality to see if it makes a true statement. If issues arise during the process, it's important to review each step for errors or consider a different approach to the problem.
Step-by-step explanation:
To prove that the solution of an inequality is valid, one must follow a systematic process that begins by understanding what the inequality expresses. Inequalities, like equations, are sentences that convey relationships between quantities. They tell us how numbers or expressions compare to each other through the use of inequality symbols (e.g., <, >, ≤, ≥).
First, you should make a complete list of what is given or can be inferred from the problem. This involves identifying the knowns—the values or relationships that are presented as facts or accepted truths. Next, you will need to solve the inequality by isolating the variable on one side. This may involve adding, subtracting, multiplying, or dividing both sides by the same number—keeping in mind that if you multiply or divide by a negative number, you must reverse the inequality sign.
Once you have a potential solution, it is crucial to check that it indeed satisfies the original inequality. You do this by substituting the solution back into the original inequality to see if it produces a true statement. If the substitution results in a true statement, you have validated the solution. In cases where you have a range of possible solutions, you may need to test more than one value within the solution set or check the solution graphically.
If at any point during your solving process you encounter difficulties or inconsistencies, such as if the substitution results in a false statement, it is important to go back and review each step you took for errors. You may also need to reconsider your interpretation of the problem or the approach you have taken.
To illustrate these concepts, consider the inequality 2x + 3 > 7. You can solve for x by subtracting 3 from both sides to get 2x > 4 and then dividing both sides by 2 to find x > 2. To check if this is a valid solution, plug any number greater than 2 back into the original inequality and confirm that it makes a true statement.