Problem 1 - Algebra 1: Concepts and Skills

Question

Determine whether the following statement is true or false. Explain. A solution of a system of linear inequalities is an ordered pair that is a solution of any one of the inequalities in the system.

Step-by-Step Solution

Answer

The statement is false. A solution of a system of linear inequalities is an ordered pair that satisfies all the inequalities in the system, not just any one of them.

Step 1: Concept Understanding

A system of linear inequalities is a set of two or more inequalities with the same variables. The solutions to the system are the coordinates (ordered pairs) that satisfy all the inequalities in the system at the same time.

Step 2: Analyzing the Statement

The statement given claims a solution of a system of linear inequalities is an ordered pair that is a solution of any one of the inequalities in the system. This means that as long as the ordered pair satisfies one of the inequalities, it would be considered a solution to the whole system. However, that contradicts with our understanding of the concept where it's required for the solution to satisfy all inequalities at once.

Step 3: Conclusion

Thus, the original statement is not true. An ordered pair is only a solution to the entire system of linear inequalities if it satisfies all the inequalities in the system simultaneously.

Key Concepts

Solution of Inequalities

A system of linear inequalities consists of multiple inequalities that need to be solved together. Each inequality narrows down the possible solutions, and the ultimate solution is where these sets of solutions overlap. It requires finding ordered pairs - points in a coordinate plane that satisfy all the inequalities. This means each one of the inequalities must be considered, and only when an ordered pair works for all these inequalities, is it considered as an acceptable solution.

For example, if you have two inequalities, each depicting a part of the coordinate plane, the solution of these inequalities would be the area where these two parts overlap. This area would contain all the ordered pairs that solve both inequalities simultaneously. This is essential because solving just one inequality might mislead you into thinking you have a complete solution, but the true solution lies in the overlapping section from all inequalities.

Ordered Pairs

Ordered pairs are essentially coordinates - values that show a point’s location on a graph. In terms of linear inequalities, an ordered pair is typically represented as \((x, y)\),where "x" is the horizontal coordinate, and "y" is the vertical coordinate.

When solving a system of linear inequalities, these pairs are evaluated to see if they satisfy all the inequalities involved. If a system has \(x > 1\) and \(y < 3\),an ordered pair \((2, 2)\)would satisfy both because \(2 > 1\)and \(2 < 3\).Therefore, \((2, 2)\)is considered a solution to this inequality system.

Ordered pairs help visualize the solutions on a graph, providing a clearer understanding of what values work for the given inequalities.

Coordinate System

A coordinate system represents a space for plotting and analyzing ordered pairs. In the context of linear inequalities, the Cartesian coordinate system is used, consisting of two perpendicular lines: the horizontal \(x\)-axisand the vertical \(y\)-axis.These intersect at the origin \((0, 0)\).

This system allows you to graphically represent inequalities and the regions they occupy. Graphing inequalities involves shading areas of the coordinate plane to indicate where the solutions lie.

• For inequalities like \(x > a\),you shade to the right of a vertical line at \(x = a\).
• Similarly, for \(y < b\),you shade below the horizontal line at \(y = b\).

The intersection of these shaded regions represents the set of solutions that satisfy the entire system of inequalities.

Understanding the coordinate system is crucial as it provides the visual framework necessary to determine where ordered pairs fit into the solution of a system of linear inequalities.