Problem 56 - 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. $$ 72 \div t>6 ; t=12 $$

Step-by-Step Solution

Answer

\(t = 12\) is not a solution to the inequality \(72 \div t > 6\).

Step 1: Substitute the given value

Substitute the given value of \(t = 12\) into the inequality \(72 \div t > 6\). It gives us \(72 \div 12 > 6\)

Step 2: Simplify the inequality

Simplify the left side of the inequality. It results in \(6 > 6\)

Step 3: Check the inequality

Check the inequality. It is not correct because 6 is not greater than 6. Therefore, \(t = 12\) is not a solution to the inequality \(72 \div t > 6\)

Key Concepts

Checking Solutions of Inequalities

Understanding how to verify the solutions to algebraic inequalities is an essential skill in algebra. The process essentially checks the validity of a proposed solution by seeing if it satisfies the inequality when substituted in.

For example, imagine you're presented with the inequality \[ 72 \div t > 6 \] and asked to determine if \( t = 12 \) is a solution. To check, you simply substitute \( t \) with 12 and evaluate it: \[ 72 \div 12 > 6 \]. This simplifies to \[ 6 > 6 \], which is not true since 6 is not greater than 6. Therefore, \( t = 12 \) is not a solution to the inequality. Students should remember that checking solutions is a way to validate or invalidate their answers, and it helps in understanding the relationships between numbers within the context of inequalities.

Substituting Values in Inequalities

Substituting values in inequalities is a process similar to that in equations. However, a critical thing to remember with inequalities is the direction of the inequality sign. When substituting values, you take the given or assumed value of the variable and replace the variable with this value in the inequality.

For instance, given the inequality and a proposed variable value, like in the exercise \( 72 \div t > 6 \) with \( t=12 \), you'd perform the substitution directly: replace \( t \) with 12 and calculate the result. It's crucial to follow the order of operations and evaluate any arithmetic carefully. Incorrect substitution or arithmetic can lead to an erroneous conclusion about the solution's validity.

Simplifying Algebraic Inequalities

Simplifying algebraic inequalities involves consolidating the inequality to its simplest form to facilitate easier analysis or further operations. This may include performing arithmetic operations, combining like terms, or factoring expressions.

In our initial example, simplifying the inequality after substitution means carrying out the division on the left side of the inequality sign: \[ 72 \div 12 = 6 \]. The inequality simplifies to \( 6 > 6 \), which indicates simplification has been done correctly. However, the resulting statement is false, thus proving the solution invalid. Simplification is a crucial step in solving and checking inequalities as it brings out the true nature of the inequality for further inspection.