Problem 49 - Algebra 1

Question

CHECKING SOLUTIONS OF INEQUALTTIES Check whether the given number is a solution of the inequality. $$y^{3}-2 \leq 8 ; 2$$

Step-by-Step Solution

Answer

Yes, the number 2 is a solution of the given inequality.

Step 1: Understanding the inequality and the proposed solution

The inequality is given as \(y^{3}-2 \leq 8\). The number 2 is suggested as a possible solution.

Step 2: Substituting the proposed solution

Replace \(y\) in the inequality with the value proposed, resulting in \((2)^{3}-2 \leq 8\). This simplifies further into \(8-2 \leq 8\).

Step 3: Evaluating the inequality with the substituted value

Simplifying the inequality, it results in \(6 \leq 8\), which holds true. Therefore, the number 2 is a valid solution to the inequality as per the given criteria.

Key Concepts

Inequality Substitution

Solving inequalities often involves a technique known as 'inequality substitution'. This is the process where you take a proposed solution and substitute it directly into the inequality to verify if the solution satisfies the inequality.

For example, given an inequality like \(y^{3} - 2 \leq 8\), and a proposed solution such as 2, substitution involves replacing the variable \(y\) with 2 in the equation. Thus, the equation becomes \((2)^{3} - 2 \leq 8\) which simplifies to \(6 \leq 8\). If the resulting statement is true, as it is in this case, then we confirm that the proposed solution is indeed valid. This straightforward method is powerful for checking potential solutions one at a time, ensuring the integrity of your answers in algebra.

Algebraic Reasoning

Behind every algebraic equation and inequality lies a foundation of logic and reasoning. Algebraic reasoning is the process of understanding and manipulating algebraic expressions and equations based on mathematical principles and properties. It is crucial when you're working with inequalities, as it guides you to decide which steps to take.

For instance, it's algebraic reasoning that informs us that when we cube the number 2, we get 8, and when we subtract 2 from 8, we confirm that the result is indeed less than or equal to 8. Recognizing and applying the correct operations to both sides of an inequality, and understanding how various operations affect the inequality, requires solid algebraic reasoning.

Evaluating Expressions

Evaluating expressions is a fundamental skill in algebra that involves performing the operations included in an algebraic expression to find its value. Every time we solve an inequality, we use this concept to simplify the algebraic expression after substitution.

An example from our initial problem would be evaluating the expression \((2)^{3} - 2\). To evaluate this, we calculate the cube of 2, which is 8, and then subtract 2, leading us to 6. This mathematical simplification confirms the truth of the inequality \(6 \leq 8\) when we input the value 2. Mastering the evaluation of expressions is essential not just in checking solutions of inequalities, but in all forms of numerical problem solving.