Problem 51 - Algebra 1: Concepts and Skills

Question

CHECKING SOLUTIONS OF INEQUALITIES Check to see if the given value of the variable is or is not a solution of the inequality. $$ n-2<6 ; n=3 $$

Step-by-Step Solution

Answer

Yes, \(n=3\) is a solution to the inequality \(n-2<6\).

Step 1: Identify the given inequality and the given value of variable

The given inequality is \(n-2<6\) and the given value of the variable 'n' is 3.

Step 2: Substitute the given value in the inequality

Substitute \(n=3\) in the inequality. The inequality becomes \(3-2<6\). Simplify it to get \(1<6\).

Step 3: Verify the inequality

Now, check if the inequality \(1<6\) is true. As 1 is indeed less than 6, it means that \(n=3\) is a solution to the inequality \(n-2<6\).

Key Concepts

Check Solutions

Checking solutions is an essential part of understanding inequalities. This process verifies whether a specific value satisfies a given inequality. For instance, when we're told to check if \( n = 3 \) is a solution to the inequality \( n - 2 < 6 \), we need to test this value within the inequality.

• First, clearly understand the original inequality, \( n - 2 < 6 \).
• Next, substitute the given value into the inequality.
• Finally, simplify the expression and check if the resulting statement is true.

This particular step confirms the validity of the solution. If the simplified inequality holds true (as it does in the example where \( 1 < 6 \)), the given value is indeed a solution. This not only helps verify solutions but also strengthens understanding of how inequalities work.

Substitution Method

The substitution method is a straightforward but powerful technique used in solving equations and inequalities. When dealing with inequalities, substituting a known value of the variable helps to test if the inequality holds true.

• Start with the original inequality \( n - 2 < 6 \).
• Replace every instance of the variable with its given value, in this case, \( n = 3 \).
• Now, the inequality turns into \( 3 - 2 < 6 \).

This substitution helps transform the inequality into a numerical expression, making it easier to evaluate the result. By isolating the constants, the inequality becomes simpler to solve, clarifying whether your substituted value makes the inequality true or false.

Simplifying Expressions

Simplifying expressions involves manipulating an equation or inequality to its simplest form. This is an essential skill for both solving and understanding inequalities thoroughly. In our example, after substituting \( n = 3 \) into the inequality \( n - 2 < 6 \), we perform arithmetic operations to simplify: \( 3 - 2 \), which simplifies to \( 1 < 6 \).

• Conduct basic arithmetic operations to simplify the inequality.
• Ensure that the inequality is straightforward to interpret at the end of the simplification.
• Check that the left side remains less than the right side (or satisfies another inequality symbol).

Simplification reduces the complexity of expressions and helps reveal their true nature. This clearer view assists us not only in testing possible solutions but also in comprehending the problem at hand more deeply.