Problem 31 - Algebra 1: Concepts and Skills

Question

Check to see if the given value of the variable is or is not a solution of the inequality. $$ 12 x \leq 70 ; x=6 $$

Step-by-Step Solution

Answer

No, \(x = 6\) is not a solution to the given inequality.

Step 1: State Given Inequality and Value

The inequality is \(12x \leq 70\). We want to test if \(x = 6\) is a solution.

Step 2: Substitute the Value of Variable

Replace \(x\) in the inequality with 6. This results in: \(12*6 \leq 70\).

Step 3: Solve the Expression

Solving the left side of the inequality gives \(72 \leq 70\). This is obviously not true.

Key Concepts

Linear Inequalities

Linear inequalities are algebraic expressions that involve a linear function and a comparison operator such as \<, \leq, \>,\ or \geq\. Unlike linear equations, which show equality, linear inequalities express a relation where one side is lesser or greater than the other. For example, the inequality \(12x \leq 70\) is a linear inequality involving a single variable, \(x\), and requires finding all the values of \(x\) that make the inequality true.

In the context of this exercise, we were examining the specific validity of \(x = 6\) against the given inequality. The purpose of linear inequalities is to describe a range of possible solutions, which in graphical terms, can be represented by shading a region on one side of a line on a coordinate plane.

Solving Inequalities

Solving inequalities involves finding all values of the variable that satisfy the inequality's condition. The process is similar to solving equations but with a keen awareness of how multiplication or division by a negative number can affect the inequality sign (reversing it).

The method of solving an inequality generally involves isolating the variable on one side of the inequality sign while keeping the inequality balanced. In the example from the textbook exercise, \(12x \leq 70\), we'd typically solve for \(x\) by dividing both sides by 12. However, in the context of verification, we want to make sure our solution set is correct, so we substituted \(x\) with 6 to see if it satisfies the original inequality. The substitution led to \(72 \leq 70\), which is not true; hence, \(x = 6\) is not a solution.

Substitution Method in Inequalities

The substitution method in inequalities is a pivotal technique used to verify if a specific value is a solution to the inequality. The method involves replacing the variable with the given value and then evaluating the resulting expression to see if the inequality holds.

For instance, in our textbook exercise, we substituted \(x\) with 6, which gave us \(12 \times 6 \leq 70\). After performing the calculation on the left side, we assessed whether the resulting statement \(72 \leq 70\) was true or false. Since 72 is greater than 70, the inequality does not hold, signaling that \(x = 6\) is not a solution to the inequality. This method is extremely useful for straightforward validation of potential solutions without the need to solve the inequality fully.