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An ordered pair consists of two numbers written in a specific order, usually enclosed in parentheses. The two numbers are typically labeled as x and y. For example, in the ordered pair (5, 46), 5 is the value of x, and 46 is the value of y.
Ordered pairs are often used to represent points on a coordinate plane, where the x-value indicates the position along the horizontal axis, and the y-value indicates the position along the vertical axis.
When dealing with systems of inequalities, we check whether a given ordered pair satisfies all the inequalities. In this exercise, the given ordered pair is (5, 46). To determine if it's a solution, we substitute these values into each inequality and verify the results.

The substitution method involves replacing variables in inequalities or equations with their numerical values from an ordered pair. This allows us to check whether the pair satisfies the inequalities.
In the provided exercise, the ordered pair is (5, 46), and the inequalities are as follows:
\[ 6x + y \leq 80 \] \[ y \geq 9x \] \[ x \geq 2 \]
We substitute x = 5 and y = 46 into each inequality to see if the pair works for all of them:

• For the first inequality: \[ 6(5) + 46 \leq 80 \] which simplifies to \[ 76 \leq 80 \] - This is true.


• For the second inequality: \[ 46 \geq 9(5)\] which simplifies to \[ 46 \geq 45 \] - This is true.


• For the third inequality: \[ 5 \geq 2 \] - This is true.

Since all three statements are true, we conclude the given ordered pair (5, 46) is a solution to the system of inequalities.

In mathematics, inequalities express a relationship where one quantity is either greater than, less than, greater than or equal to, or less than or equal to another quantity.
Inequalities are crucial for defining ranges of values rather than exact numbers. For instance, the inequality \[ x \geq 2 \] tells us that x can be any number greater than or equal to 2.
In a system of inequalities, multiple inequalities are considered together to find common solutions that satisfy all conditions. Let's take a deeper look at the inequalities given in the original exercise:

• First inequality: \[ 6x + y \leq 80 \] - This means the sum of six times the x-value plus the y-value should be less than or equal to 80.


• Second inequality: \[ y \geq 9x \]- Indicates that the y-value should be at least nine times the x-value.


• Third inequality: \[ x \geq 2 \]- Implies that the x-value should be greater than or equal to 2.

For a solution to be valid, it must satisfy all these inequalities simultaneously. In our case, the ordered pair (5, 46) works since it meets all the given conditions.