Problem 52 - Algebra 1: Concepts and Skills

Question

Check whether the given value of the variable is a solution of the inequality. (Lesson 1.4) $$ 2 x<24 ; x=8 $$

Step-by-Step Solution

Answer

Yes, x = 8 is a solution to the inequality '2x < 24', since after substituting the value, the inequality '16 < 24' holds true.

Step 1: Identify the inequality and given value

The given inequality is '2x < 24' and the value of x provided is 8.

Step 2: Substitute the value into the inequality

Replace 'x' in '2x < 24' with 8. This results in the inequality '2*8 < 24'.

Step 3: Evaluate the inequality

After substitution, the inequality becomes '16 < 24'. If this statement is true, then the given value for 'x' is a solution for this inequality.

Key Concepts

Inequality Substitution

One of the essential techniques in algebra is inequality substitution, which is employed when you're given a possible solution to an inequality and asked to verify its validity. To use this technique, you simply take the value provided—often for a variable such as 'x'—and substitute it directly into the inequality in the place of the variable.

For instance, if you're given the inequality '2x < 24' and want to check if x = 8 is a solution, you would replace 'x' with 8, making the inequality '2(8) < 24'. It's a straightforward process, yet it’s crucial for efficiently checking whether values satisfy the conditions of an inequality.

Practical Application

In real-world situations such as budgeting or calculating dosages, you often need to know whether a certain number ('x') falls within an acceptable range. Substituting the number into an inequality helps in making rapid, accurate assessments.

Algebraic Inequalities

Moving beyond basic operations, algebraic inequalities serve as a critical part of understanding mathematical relationships. An inequality, unlike an equation, tells us that two expressions are not equal but rather one is greater than or less than the other. Expressions like '2x < 24' showcase this by indicating a range of values for 'x' that make the inequality true.

There are several types of inequalities including less than ('<'), greater than ('>'), less than or equal to ('≤'), and greater than or equal to ('≥'). It's important to be comfortable with these symbols since they're fundamental to expressing the results after evaluating inequalities.

Building Strong Foundations

Developing a good grasp of algebraic inequalities enables you to tackle more advanced topics in mathematics, including multi-step inequalities and systems of inequalities, with greater confidence and understanding.

Evaluating Inequalities

The last step in verifying a solution to an inequality is evaluating the inequality after substitution. This step requires you to perform any arithmetic implied by the substitution and then interpret the results to conclude whether the inequality holds true. If our substituted inequality is '16 < 24', the evaluation step confirms that 16 is indeed less than 24.

Effectively evaluating inequalities ensures that you can comprehend and prove whether a variable's value is a part of the solution set. Correct evaluation empowers you to solve more complex problems

Why Evaluation Matters

The evaluation process in inequalities is analogous to proof-checking in real-world scenarios, such as making sure a structure’s measurements meet specified safety standards. By practicing this, you develop careful analysis skills applicable in academic and practical contexts alike.