Testing Solutions to Equations and Inequalities
Students use substitution to determine whether given values make one-variable equations and inequalities true.
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Equations, Inequalities, and Solutions
An equation states that two expressions are equal. For example, x + 4 = 10 says that x + 4 and 10 have the same value. An inequality compares expressions using symbols such as <, >, ≤, or ≥. For example, x + 4 > 10 says that x + 4 is greater than 10. A solution is a value that makes the equation or inequality true. To test x = 6 in x + 4 = 10, replace x with 6. The result is 6 + 4 = 10, or 10 = 10, so 6 is a solution. For the inequality x + 4 > 10, the same value gives 10 > 10. That statement is false, so 6 is not a solution to the inequality.
Substituting a Given Value
Substitution means replacing a variable with a given number. Use parentheses when you substitute, especially when multiplication or a negative number is involved. Suppose you want to test whether n = 4 is a solution to 3n − 2 = 10. Replace n with 4 to get 3(4) − 2 = 10. Then simplify: 12 − 2 = 10, which becomes 10 = 10. Because the final statement is true, n = 4 is a solution. Be careful to replace every occurrence of the variable. For example, testing x = 3 in 2x + x = 9 gives 2(3) + 3 = 9. Both x terms must be replaced. Correct substitution lets you check a value without first solving the equation.
Evaluating Both Sides
After substituting, evaluate the left side and the right side separately. Do not assume the sides are equal just because an equal sign appears between them. Consider whether y = 5 is a solution to 2y + 1 = 3y − 4. Substitute 5 for every y. The left side becomes 2(5) + 1 = 10 + 1 = 11. The right side becomes 3(5) − 4 = 15 − 4 = 11. Since both sides have the value 11, the equation is true, and y = 5 is a solution. Follow the order of operations on each side. Simplify multiplication before addition or subtraction. Keeping the two sides separate helps prevent errors and makes the comparison easy to see.
Deciding Whether a Statement Is True
The final step is to decide whether the simplified statement is true or false. For an equation, compare the two values for equality. For an inequality, read the comparison symbol carefully. Test a = 7 in 2a − 3 ≤ 10. Substitution gives 2(7) − 3 ≤ 10. Simplifying produces 14 − 3 ≤ 10, or 11 ≤ 10. This is false because 11 is not less than or equal to 10. Therefore, a = 7 is not a solution. If a = 6 is tested instead, the result is 9 ≤ 10, which is true, so 6 is a solution. A false result does not mean your work failed; it correctly shows that the tested value is not in the solution set.
Guided Practice
Test each given value by substituting, simplifying both sides, and naming the result. First, test m = 3 in 4m + 2 = 14. Substitution gives 4(3) + 2 = 14, and simplification gives 14 = 14. The statement is true, so 3 is a solution. Next, test p = 5 in 2p − 1 > 8. The result is 2(5) − 1 > 8, then 9 > 8. This is true, so 5 is a solution. Finally, test k = 2 in 5 + k ≥ 8. Substitution gives 5 + 2 ≥ 8, or 7 ≥ 8. This statement is false, so 2 is not a solution. Always include a final sentence that connects the truth value to the tested number.
Independent Solution Check
Use a consistent routine when checking a solution independently. Write the original equation or inequality, replace every variable with the given value, evaluate each side, decide whether the result is true, and state your conclusion. For example, check whether r = −2 is a solution to 3r + 4 < 0. Substitute using parentheses: 3(−2) + 4 < 0. Simplify to −6 + 4 < 0, then −2 < 0. The statement is true, so r = −2 is a solution. Parentheses are especially helpful with negative values because they show that the entire number replaces the variable. Before finishing, check the comparison symbol and your arithmetic. Your conclusion should say either that the given value is a solution or that it is not a solution.
