Question

CHECKING SOLUTIONS OF INEQUALTTIES Check whether the given number is a solution of the inequality. $$\frac{25-d}{d} \geq 4 ; 5$$

Short answer

Yes, 5 is a solution to the inequality \(\frac{25-d}{d} \geq 4 \).

Step-by-step solution

Identify Given Number and Inequality

The given number is 5 and the inequality to check is \(\frac{25-d}{d} \geq 4 \).

Substitute Given Number Into the Inequality

We need to substitute d with 5 in the inequality \(\frac{25-d}{d} \geq 4 \) which results in \(\frac{25-5}{5} \geq 4 \), simplify this to \(\frac{20}{5} \geq 4 \).

Evaluate and Check if the Inequality Holds

We evaluate \(\frac{20}{5} \) to get 4, so we have \(4 \geq 4\). This is a true statement, meaning that 5 is indeed a solution to the inequality.

Key concepts

Inequality

In the realm of mathematics, an inequality is a statement about the relative size or order of two objects, or about whether they are the same or not. In other words, it shows how one number or expression is greater than, less than, or equal to another number or expression.

An inequality is expressed using symbols such as \textgreater, \textless, \textgreater= (greater than or equal to), and \textless= (less than or equal to). These are fundamental in various mathematical analyses and real-world applications, where relationships between quantities are not exact but have upper or lower limits.

For instance, considering the basic inequality \textless=, like in the statement \( x \textless= 10 \), it is understood that x could be the number 10 or any number less than 10. This gives a range of possible values for x, as opposed to a strict equality which would limit x to a single value.

Substitute in Inequality

When presented with an inequality that contains a variable, one common task is to substitute a specific value in place of the variable to determine if the inequality holds true or false. This process involves replacing the variable with a given number and simplifying the expression as far as possible.

To substitute correctly, it's important to follow the order of operations and use parentheses where necessary to avoid errors. For example, if we have the inequality \( 3x + 2 \textgreater 5 \) and we want to check if x = 4 is a solution, we substitute 4 in for x to get \( 3(4) + 2 \textgreater 5 \), which simplifies to \( 12 + 2 \textgreater 5 \). It is clear that \( 14 \textgreater 5 \), thus confirming that x = 4 is indeed a solution to this inequality.

Evaluate Inequality

To evaluate an inequality means to perform the necessary arithmetic operations to determine the truth value of the statement. This involves carrying out any addition, subtraction, multiplication, or division as indicated and then comparing the resulting expressions based on the initial inequality sign.

The outcome of this process will tell us whether we have a true or false statement. If the initial inequality sign remains appropriate after the evaluation, then the statement is true. Otherwise, if the result contradicts the initial inequality sign, the statement is false. For example, after substituting a value into the inequality \( \frac{25-d}{d} \textgreater = 4 \), evaluating it by carrying out the division and simplification will lead us to a clear conclusion about the veracity of the inequality given the substituted value.