Question

Check whether the given number is a solution of the inequality. $$x+5>11 ; 6$$

Short answer

No, 6 is not a solution of the given inequality.

Step-by-step solution

Understand the inequality

The inequality \(x + 5 > 11\) means that when a certain number is added to 5, the result should be greater than 11.

Substitute the given number

Now, replace \(x\) in the inequality with the given number, which is 6 in this case. So, it becomes \(6 + 5 > 11\).

Solve the inequality

Do the addition on the left side of the inequality: \(11 > 11\).

Verify the result

Now, the inequality \(11 >11\), is not true because 11 is not greater than 11. Therefore, the given number 6 is not a solution for the inequality.

Key concepts

Inequality Substitution

When faced with an inequality, such as \(x + 5 > 11\), one way to determine if a particular number is a solution is to perform inequality substitution. This method involves taking the original inequality and replacing the variable with the given number, much like plugging in values to check solutions in an equation.

For example, if the number 6 is to be checked against the given inequality, the substitution process would replace every instance of \(x\) with 6, which transforms the original inequality into \(6 + 5 > 11\). This new expression can be simplified to \(11 > 11\), setting the stage for the next step in our verification process. Substitution is a cornerstone of algebraic problem-solving, as it provides a straightforward way to test different values against an inequality’s condition.

Inequality Verification

After substituting a value into our inequality, it’s crucial to undergo inequality verification. This step involves checking if the result after substitution maintains the truth of the original inequality. In our example, substituting 6 into the inequality resulted in \(11 > 11\). Upon inspection, this statement is not true because 11 is not greater than 11, but rather equal to it. This means the number 6 does not satisfy the condition that the sum of \(x\) and 5 must be greater than 11.

Verification is essential because it confirms the mathematical validity of the solution within its context. Without this step, it's easy to overlook crucial aspects of inequality solutions, such as understanding the relationships between 'greater than', 'less than', 'greater than or equal to', and 'less than or equal to'.

Algebraic Reasoning

At the heart of solving inequalities is algebraic reasoning, which is the process of forming logical conclusions based on the properties and operations of algebra. For example, understanding that adding the same number to both sides of an inequality maintains the inequality's direction is part of algebraic reasoning. When approaching an inequality like \(x + 5 > 11\), application of this reasoning is essential.

With the given inequality, you would deduce that any number greater than 6, when replaced for \(x\), would make the inequality true. This is because if \(x = 6\), it yields \(11 > 11\), which is on the cusp of being true—if \(x\) were any larger, the left side of the inequality would indeed be greater than the right. In summary, algebraic reasoning involves understanding these kinds of relationships and operations to solve for variables efficiently and correctly.