Problem 51 - Algebra 1

Question

Check whether the given number is a solution of the inequality. $$4+7 y>12 ; 1$$

Step-by-Step Solution

Answer

No, 1 is not a solution to the inequality \(4 + 7y > 12\).

Step 1: Understand the Inequality

In the inequality \(4 + 7y > 12\), we are saying that whatever value 'y' takes, once multiplied by 7 and added to 4, it should give a result that is greater than 12.

Step 2: Substituting the given value

Now, we have to check whether 'y = 1' is a solution for our inequality. We do this by substituting y = 1 into the inequality like this: \(4 + 7(1)\).

Step 3: Simplifying the Inequality

If we simplify the inequality, we get \(4+7 = 11\).

Step 4: Comparing the result

Now, we need to compare our result with the right-hand side of the inequality. If it's larger than 12, then y = 1 is a solution. However, 11 is not greater than 12.

Key Concepts

Inequality Substitution

When solving mathematical inequalities, substitution is a critical step in determining if a particular value satisfies the inequality.
Essentially, inequality substitution involves replacing the variable in an inequality with a given number to see if the inequality holds true. For example, when given an inequality such as \(4 + 7y > 12\), and asked to check if \(y = 1\) is a solution, you directly substitute 1 for y and assess the result.

This process is straightforward, yet it serves as the foundation for testing solutions to inequalities. It’s crucial to substitute correctly and simplify the expression to get an accurate comparison with the other side of the inequality, which ultimately informs us if our initial value is indeed a solution.

Algebraic Expressions

At the heart of solving inequalities lies a firm understanding of algebraic expressions. An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. In the context of our example inequality \(4 + 7y > 12\), \(4 + 7y\) is an algebraic expression.

Expressions become especially important when assessing inequalities because they represent the variable part of the inequality. Learning to manipulate these expressions—such as combining like terms or distributing factors—is essential in the simplification process and is a steppingstone for solving complex inequalities.

Mathematical Inequalities

Mathematical inequalities are statements about the relative size or order of two quantities. They are a way to express that one quantity is greater than, less than, equal to, or not equal to another quantity. Inequalities are depicted with signs like '>', '<', '\(\geq\)', and '\(\leq\)'.

In practice, solving inequalities such as \(4 + 7y > 12\) involves finding all possible values of y that make the inequality true. These solutions could be a single number, a range of numbers, or sometimes even no solution at all. Understanding how to correctly interpret and manipulate these inequality signs is integral to finding the correct solution set.

Inequality Simplification

Inequality simplification is the process of breaking down complex expressions within an inequality into simpler, more manageable parts without changing the inequalities' original meaning. This involves applying basic arithmetic operations and properties of equality to both sides of the inequality.

The goal of simplification is to isolate the variable, making it easier to compare it with the other side of the inequality. For instance, the expression \(4 + 7y\) simplifies to 11 when \(y = 1\). Simplification enables us to clearly see that 11 is not greater than 12, therefore, \(y = 1\) is not a solution to the inequality \(4 + 7y > 12\). A systematic approach to simplification is fundamental in solving inequalities, as it can lead to a clear deduction about the nature of the solution.