Problem 71 - Algebra 1

Question

Check whether the given number is a solution of the inequality. $$ 2 x<24 ; 8 $$

Step-by-Step Solution

Answer

Yes, 8 is a solution to the given inequality.

Step 1: Substitute the Value

Substitute the value 8 for x in the inequality. \(2*8 < 24.\)

Step 2: Perform the Calculation

Perform the multiplication on the left side of the inequality. \(16 < 24.\)

Step 3: Verify the Inequality

Check if the resulting expression \(16 < 24\) is true. As 16 is indeed less than 24, this verifies that 8 is a solution to the given inequality.

Key Concepts

Solution Verification

Solution verification is an essential step in solving inequalities because it allows us to confirm whether a given number truly satisfies the inequality. This process ensures that our mathematical work is accurate and reliable.
To verify a solution, we substitute the proposed number into the inequality in place of the variable, then simplify the expression.
The result should maintain the truth of the inequality.

• If substituting the number leads to a valid inequality (e.g., "less than" remains true), then the proposed number is indeed a solution.
• Conversely, if the inequality becomes false or invalid, the number cannot be considered a solution.

Being diligent in solution verification helps in eliminating errors and achieving confidence in problem-solving, solidifying one's understanding of inequalities.

Mathematical Substitution

Mathematical substitution is a straightforward yet powerful tool in solving mathematical expressions including inequalities. In the context of substitution, we replace a variable with a provided number to test or evaluate an expression or inequality.
This technique is particularly useful as it allows us to simplify the problem and focus on solving it step by step.

• For inequalities, substitution lets us see if a specific value maintains the truth of the expression.
• Substitution helps break down complex equations into smaller, more manageable pieces.

In our example, substituting 8 for \(x\) in the inequality \(2x < 24\) converts it into \(2 \times 8 < 24\). This simple change greatly assists in subsequent calculations and solution verification.

Algebraic Expressions

Algebraic expressions form the basis of algebra, incorporating numbers, variables, and operations such as addition, subtraction, multiplication, and division. Understanding these expressions is crucial when dealing with inequalities, where expressions are compared using "less than," "greater than," and other relational signs.
An important aspect of working with algebraic expressions is being able to manipulate and simplify them without changing their fundamental relationships.

• In inequalities, algebraic expressions help define the conditions we are examining—either a number is a solution, or it isn’t.
• Solving inequalities involves finding the set of numbers that make the expression true, requiring a good grasp of how algebraic structures and operations interact.

Mastering algebraic expressions helps in efficiently tackling inequalities and many other branches of mathematics, paving the way to better problem-solving and critical thinking skills.