Problem 52 - Algebra 1

Question

Check whether the given number is a solution of the inequality. $$3 x-3 \geq x+3 ; 5$$

Step-by-Step Solution

Answer

Yes, 5 is a solution to the inequality \(3x - 3 \geq x + 3\).

Step 1: Substitute the given number into the inequality

Begin by substituting the given number (5) into the inequality \(3x - 3 \geq x + 3\) so that it becomes \(3*5 - 3 \geq 5 + 3\).

Step 2: Simplify the inequality

Next step is to simplify the inequality. Here, \(3*5 - 3\) simplifies to \(12\) and \(5 + 3\) simplifies to \(8\). Our inequality thus becomes \(12 \geq 8\).

Step 3: Check if the resulting inequality is true

After we simplify, we need to check if the inequality is true or not. From the previous step, we have \(12 \geq 8\), and because \(12\) is indeed greater than or equal to \(8\), this inequality is true.

Key Concepts

Substitution Method

The substitution method is an essential tool for testing potential solutions to inequalities. It involves replacing the variable in an inequality with a specific number to determine whether the inequality holds true.

For example, let's consider the inequality \(3x - 3 \geq x + 3\). To use the substitution method, we take the given candidate solution, which in this case is the number 5, and substitute it in place of \(x\). Thus, \(3*5 - 3\) replaces \(3x - 3\) and \(5 + 3\) replaces \(x + 3\). After the substitution, we can simplify and evaluate the truth of the inequality.

Inequality Simplification

Simplifying an inequality is a crucial step to find out whether the potential solution is correct or not. This process entails performing arithmetic operations to reduce the expression to a more manageable form.

In our example, after substitution, we have \(3*5 - 3 \geq 5 + 3\). This simplifies to \(15 - 3 \geq 8\) and further to \(12 \geq 8\). This step is vital as it makes it very clear whether the inequality is true or false, by comparing two simple numbers.

Simplification often involves:

• Distributing multiplication over addition and subtraction
• Combining like terms
• Isolating the variable on one side when searching for the variable's possible range of values

Validating Solutions

Once simplification is completed, validating the solution is the final step. This step is about checking whether the simplified inequality accurately represents a true statement.

In our example, the simplified inequality is \(12 \geq 8\). Here, we clearly see that 12 is greater than 8, which makes the inequality true. Therefore, we can confidently say that 5 is a solution to the original inequality \(3x - 3 \geq x + 3\).

Validation helps us confirm that not only did our algebraic manipulations maintain the integrity of the inequality, but also that the potential solution meets the criterion set forth by the original inequality.