Problem 48 - Algebra 1

Question

CHECKING SOLUTIONS OF INEQUALTTIES Check whether the given number is a solution of the inequality. $$6 x-16<20 ; 7$$

Step-by-Step Solution

Answer

After substitution, we get \(42 - 16 < 20\) which simplifies to \(26 < 20\). Therefore, 7 is not a solution to the inequality.

Step 1: Substitution

Replace \(x\) in the inequality \(6x - 16 < 20\) with 7.

Step 2: Simplify

Simplify the left side of the inequality by multiplying 6 and 7, and then subtracting 16.

Step 3: Compare

Finally compare the result from step 2 with 20 to check if the inequality holds true.

Key Concepts

Inequality Solution Verification

Understanding how to verify the solutions to inequalities is an essential skill in algebra. It empowers students to confirm if their answers are correct, and it builds a strong foundation for solving more complex problems. When verifying a solution, the goal is to check whether the given number satisfies the inequality when substituted for the variable. If the inequality remains true after the substitution, the number is indeed a solution. For example, when given the inequality \(6x - 16 < 20\) and the number 7, we substitute 7 for \(x\) and evaluate. If after substitution the statement is true, then 7 is a solution to the inequality.
To ensure understanding, it's helpful to imagine this process as a 'truth test' for the inequality. If the statement remains true, you've passed the test! Employing this verification process is a vital step and enhances confidence in the accuracy of one's work.

Substitution Method

The substitution method is a straightforward yet powerful technique used in mathematics to determine whether a particular value is a solution to an inequality or equation. It involves replacing the variable in the inequality with a specific number and simplifying the expression to see if the inequality holds true. In our example, substituting the variable \(x\) with 7 transforms the original inequality \(6x - 16 < 20\) into \(6(7) - 16 < 20\).
With the substitution made, the next step is to perform the necessary arithmetic. Multiplication and subtraction will simplify the expression which can then be compared against the other side of the inequality. When teaching substitution, it's important to emphasize the accuracy of operations and maintaining equal value on both sides. This method is not only used for checking solutions but also in solving systems of equations, a topic that students will encounter frequently.

Inequality Simplification

Inequality simplification involves reducing an inequality to its simplest form to make it easier to interpret and solve. It is a critical step after substitution, as it allows students to clearly see whether the inequality is true or false. Simplification may include arithmetic operations such as addition, subtraction, multiplication, and division. In the context of our example, once we've substituted 7 into \(6x - 16 < 20\), simplifying this expression by carrying out the multiplication \(6 \times 7\), and then the subtraction \(- 16\), we arrive at \(42 - 16 < 20\), which further simplifies to \(26 < 20\). If the resulting statement is false, as in this case, then the original inequality does not hold true for the number we substituted.
Students should grasp that the simplification process is like peeling back layers to reveal the core truth of the inequality. It's essential to perform each operation carefully and in the correct sequence to avoid errors. Simplification paves the way for a clear comparison between the two sides of the inequality, leading to an accurate verification of the solution.