Question

Is each ordered pair a solution of the inequality? $$1.8 x-3.8 y \geq 5 ;(0,0),(1,-1)$$

Short answer

The ordered pair (0,0) is not a solution to the inequality, but the ordered pair (1,-1) is a solution to the inequality.

Step-by-step solution

Substitute the first ordered pair

Throughout this step, the first ordered pair (0,0) should be replaced for x and y in the given inequality, which is \(1.8 x-3.8 y \geq 5\). The inequality becomes \(1.8 * 0 - 3.8 * 0 \geq 5\), simplifying to \(0 \geq 5\).

Evaluate the first substitution

Now, assess whether the resulting inequality from the substitution in step one holds. It is clear that \(0 \geq 5\) is false, hence, the ordered pair (0,0) is not a solution to the given inequality.

Substitute the second ordered pair

In this step, the second ordered pair (1,-1) should be replaced for x and y in the inequality \(1.8 x -3.8 y \geq 5\). This turns the inequality into \(1.8 * 1 - 3.8 * -1 \geq 5\), which simplifies to \(5.6 \geq 5\).

Evaluate the second substitution

The inequality made from the second substitution is examined now. Since \(5.6 \geq 5\) is true, it is confirmed that the ordered pair (1,-1) is a solution to the inequality \(1.8 x - 3.8 y \geq 5\).

Key concepts

Ordered Pairs

In the context of algebra, an ordered pair is a pair of numbers used to indicate the position of a point on a coordinate plane. The ordered pair is written in the form \( (x, y) \), where \( x \) is the horizontal value (also called the x-coordinate or abscissa), and \( y \) is the vertical value (y-coordinate or ordinate). These pairs are fundamental in graphing equations and inequalities.

When determining if an ordered pair is a solution for an inequality, like in our exercise \( 1.8 x - 3.8 y \geq 5 \), you essentially plug the values of the ordered pair into the inequality in place of \( x \) and \( y \) to check if the inequality holds true. If the result is true, then the ordered pair is a solution to the inequality.

Substitution Method

The substitution method is a technique used to solve systems of equations, as well as to verify if an ordered pair is a solution for an equation or inequality. This method involves replacing variables with their corresponding values from an ordered pair or another equation.

In our example, the substitution method is applied by replacing \( x \) and \( y \) with the values given by the ordered pairs \( (0,0) \) and \( (1,-1) \). This process transforms the algebraic expression into a numerical statement that can be evaluated as true or false. For a thorough understanding, it's important to perform each substitution step carefully to avoid errors that might lead to incorrect conclusions about the solutions.

Inequality Solutions

Determining inequality solutions involves finding all the ordered pairs that satisfy the inequality. An inequality like \( 1.8 x - 3.8 y \geq 5 \) specifies a set of points that forms a region on the coordinate plane. To verify individual solutions, as we did in our exercise, you use the substitution method to check if the ordered pairs make the inequality true.

For the ordered pair \( (0,0) \) the substitution didn't satisfy the inequality, making it not a part of the solution set. However, substituting \( (1,-1) \) resulted in a true statement, hence, it is included in the solution set. Understanding the concept of inequalities extends to recognizing regions, graphing, and analyzing the solutions visually as well, which are all critical skills in algebra.