Problem 46 - Algebra 1

Question

CHECKING SOLUTIONS OF INEQUALTTIES Check whether the given number is a solution of the inequality. $$t^{2}+6>40 ; 6$$

Step-by-Step Solution

Answer

Yes, the number 6 is a solution of the inequality \(t^{2}+6>40\)

Step 1: Understand the Inequality

The inequality provided is \(t^{2}+6 > 40\). This inequality means that for a number t, when it is squared and added to 6, the value obtained should be greater than 40.

Step 2: Substitution

Replace t in the inequality with the given number which is 6. This substitution gives us \(6^{2}+6 > 40\).

Step 3: Simplify the Left-Hand Side

Next, we simplify the left side of the inequality. Squaring 6 gives 36 and adding 6 gives us 42. So the inequality becomes \(42 > 40\).

Step 4: Compare

In this final step, we check if the left side of the inequality is indeed larger than the right side. As 42 is indeed larger than 40, the inequality \(42 > 40\) is true.

Key Concepts

Inequality Substitution

Inequality substitution is a technique that allows us to determine whether a specific value satisfies a given inequality. This process involves replacing the variable in the inequality with the given number and then evaluating the result. For instance, consider the inequality t^2 + 6 > 40, and we want to check if t = 6 is a solution.

We substitute 6 for t in the inequality, resulting in 6^2 + 6 > 40. This step essentially turns the inequality into an arithmetic problem that we can solve. By substitution, we transform an abstract inequality into a definite comparison between numbers. It's like trying on a key to see if it fits a lock — the substitution either confirms or denies that we have a solution on our hands. It’s crucial to replace the variable consistently throughout the inequality to avoid any errors in evaluation.

Inequality Simplification

Once substitution is done, we move to inequality simplification, where the goal is to break down the inequality into its simplest form to facilitate easy evaluation. Simplification often involves arithmetic operations like addition, subtraction, multiplication, or division.

Using the example mentioned earlier, after substituting 6 into the inequality, we get 6^2 + 6 > 40. To simplify, we calculate 6^2 which equals 36, and then add 6 to get 42. This gives us a simplified inequality 42 > 40. Simplification helps to clear up any complexities and puts the inequality into an easily comparable format. This process is critical in understanding the true nature of the inequality and preparing it for the final step of evaluation.

Evaluating Inequalities

Evaluating inequalities is the final step to confirm whether the substitution and simplification yield a true statement. In this stage, we compare the simplified expression with the other side of the inequality.

Referring back to our original example, we have simplified the inequality to 42 > 40. Now it's about assessing whether this is a true statement. Here, it is clear that 42 is indeed greater than 40, so the original inequality t^2 + 6 > 40 is true when t = 6. If the comparison doesn't hold up—that is, if the left-hand side is not greater than the right-hand side for a '>' inequality—the number in question is not a solution. The evaluation confirms the validity of our number and whether it does indeed satisfy the original inequality.