Problem 50 - Algebra 1

Question

Check whether the given number is a solution of the inequality. $$y-5>4 ; 9$$

Step-by-Step Solution

Answer

No, the number '9' is not a solution to the inequality \(y-5 > 4\).

Step 1: Understanding the inequality

The given inequality is \(y - 5 > 4\). This inequality is expressing that the value of 'y', when 5 is subtracted from it, should be greater than 4.

Step 2: Incorporating the given value

The number provided to check against the solution is '9'. Hence replace 'y' in the inequality with '9': \(9 - 5 > 4\).

Step 3: Evaluating the inequality with the given number

Evaluate the left side of the inequality: \(9 - 5 = 4\). So, inequality becomes: \(4 > 4\).

Step 4: Validating the inequality

As per the definition of 'greater than', one quantity must be more than the other quantity. But in the evaluated inequality \(4 > 4\), left side is not greater than the right side. Hence the inequality does not hold true.

Key Concepts

Algebraic Inequalities

Algebraic inequalities are akin to scales that need balancing, with a crucial difference: rather than aiming for equality, they express relationships of greater than or less than between algebraic expressions. In the problem given, we're presented with an inequality, which is a statement that one quantity is larger (or smaller) than another, notated with symbols like '>', '<', 'geq', or 'leq'. The inequality in question, y - 5 > 4, establishes that whatever value 'y' might take, once we subtract 5 from it, it must be larger than 4 for the inequality to hold true.

Inequalities are fundamental in expressing ranges of solutions, setting constraints in real-world problems, and revealing intervals of possibilities. It's important to note that inequalities have some unique properties compared to equalities, especially when it comes to operations like multiplication and division by negative numbers, which flip the inequality sign. Understanding the underlying concept of algebraic inequalities ensures a strong foundation for solving them accurately.

Evaluating Expressions

Evaluating expressions is a critical aspect of algebra that involves substituting numbers into an expression in place of variables and then simplifying. The provided exercise required evaluating the expression y - 5 by replacing 'y' with 9, leading to 9 - 5. Evaluating the numbers results in 4, setting the stage to compare whether this result is indeed greater than 4, as the original inequality demands.

Evaluating expressions is not only about substitution and simplification but also about understanding the properties and operations governing algebraic entities. Consistent practice in evaluating expressions is instrumental in achieving proficiency with algebraic manipulation and setting forth towards more complex problem-solving scenarios in mathematics.

Inequality Solutions

Inequality solutions involve finding the values that satisfy the inequality condition. These solutions can be a range of numbers, and they are often represented on a number line or in interval notation. In the exercise, the evaluation process resulted in an inequality 4 > 4, which is not true because 4 is not greater than itself. Thus, the number 9 is not a solution to the inequality y - 5 > 4.

Finding solutions to inequalities requires a clear understanding of comparison symbols and the meanings they convey. Unlike equations which usually have a single solution or a set of discrete solutions, inequalities often have infinite solutions represented in the form of intervals. It's crucial to not only find the solutions but also understand how to properly express and graph these solution sets as part of the complete resolution process.