Problem 58 - Algebra 1

Question

Check whether the number is a solution of the equation or the inequality. \(n+2 \leq 2 n-2 ; 4\)

Step-by-Step Solution

Answer

Yes, the number \(4\) is a solution to the given inequality \(n+2 \leq 2 n-2\).

Step 1: Substitute the given number into the equation

Replace all instances of \(n\) in the inequality with \(4\), the number that we are checking. The inequality now becomes \(4 + 2 \leq 2 * 4 - 2\).

Step 2: Simplify both sides of the inequality

On the left side, \(4 + 2\) simplifies to \(6\). On the right side, \(2 * 4 - 2\) simplifies to \(6\). Therefore, the inequality now is \(6 \leq 6\).

Step 3: Verify the inequality

In this case, we see that both sides of the inequality are equal, which means the inequality is true. Therefore, \(4\) is a solution for the inequality \(n+2 \leq 2 n-2\).

Key Concepts

Solution Verification

Solution verification is all about ensuring that a number truly satisfies a given equation or inequality. It involves checking if plugging a particular value into an equation makes it hold true. For instance, in the inequality \( n + 2 \leq 2n - 2 \), checking if \( n = 4 \) is a solution involves substituting \( 4 \) into the inequality and verifying its truth.

• This step confirms if your solution is accurate and valid.
• If the result is true after substitution, the number is a solution.
• Conversely, if it's false, you've likely either selected or calculated an incorrect solution.

After verification, we confirmed that the inequality holds true when \( n = 4 \), thus verifying \( 4 \) as a valid solution.

Substitution Method

The substitution method is a technique often used in solving equations and inequalities. This involves replacing the variable in an equation with a given value, which helps in determining if the chosen number can solve the equation or inequality.In our example:

• We replaced \( n \) with \( 4 \) because we wanted to see if \( 4 \) is a solution.
• In the inequality, \( n + 2 \leq 2n - 2 \), this meant replacing every \( n \) with \( 4 \).
• This gives us the inequality: \( 4 + 2 \leq 2 \times 4 - 2 \).

It's crucial to carefully substitute and simplify, as any small mistake can lead to incorrect conclusions about the solution.

Simplifying Expressions

Simplifying expressions is a key step in solving equations or inequalities, particularly after substituting a value. It involves breaking down complex expressions to make them easier to handle and solve.In our example, after substitution:

• On the left side, \( 4 + 2 \) simplifies to \( 6 \).
• On the right side, \( 2 \times 4 - 2 \) also simplifies to \( 6 \).

Now, the inequality simplifies to \( 6 \leq 6 \). By simplifying, you clearly see if both sides of the equation match, confirming if the given number is a solution. Simplification helps identify the truth of an equation, making problem-solving more intuitive and manageable.