Problem 281 - Elementary Algebra

Question

In the following exercises, determine whether each ordered pair is a solution to the system. \(\left\\{\begin{array}{l}2 x+3 y \geq 2 \\ 4 x-6 y<-1\end{array}\right.\) (a) \(\left(\frac{3}{2}, \frac{4}{3}\right)\) (b) \(\left(\frac{1}{4}, \frac{7}{6}\right)\)

Step-by-Step Solution

Answer

Pair (a) is not a solution. Pair (b) is a solution.

Step 1: Test the first inequality for pair (a)

Substitute the values from the ordered pair \(\frac{3}{2}, \frac{4}{3}\) into the first inequality. Thus, we get \2\frac{3}{2} + 3\frac{4}{3} \geq 2. This simplifies to 3 + 4 \geq 2, which is true since 7 \geq 2.

Step 2: Test the second inequality for pair (a)

Substitute the values from the ordered pair \(\frac{3}{2}, \frac{4}{3}\) into the second inequality, resulting in \4\frac{3}{2} - 6\frac{4}{3} <-1. This simplifies to 6 - 8 <-1. This is false since -2 is not less than -1.

Step 3: Conclusion for pair (a)

Since the ordered pair \(\frac{3}{2}, \frac{4}{3}\) does not satisfy the second inequality, it is not a solution to the system.

Step 4: Test the first inequality for pair (b)

Substitute the values from the ordered pair \(\frac{1}{4}, \frac{7}{6}\) into the first inequality. Thus, we get \2\frac{1}{4} + 3\frac{7}{6} \geq 2. This simplifies to \frac{1}{2} + \frac{7}{2} \geq 2, which is true since \4 > 2.

Step 5: Test the second inequality for pair (b)

Substitute the values from the ordered pair \(\frac{1}{4}, \frac{7}{6}\) into the second inequality, resulting in \4\frac{1}{4} - 6\frac{7}{6}<-1. This simplifies to 1 - 7 \ < -1. This is true since -6 \ < -1.

Step 6: Conclusion for pair (b)

Since the ordered pair \(\frac{1}{4}, \frac{7}{6}\) satisfies both inequalities, it is a solution to the system.

Key Concepts

Ordered Pairs

An ordered pair is a pair of numbers used to locate a point on a coordinate plane.
It is typically written in the form \(x, y\) where 'x' represents the horizontal position and 'y' represents the vertical position.
In the context of systems of inequalities, an ordered pair is considered a solution if it satisfies all inequalities in the system.
This means you will substitute the 'x' and 'y' values of the ordered pair into each inequality and see if the statement is true.
If the ordered pair makes all the inequalities true, then it is a solution to the system.

Substitution Method

The substitution method is a technique used to solve systems of equations and inequalities.
It involves substituting the values of the variables from an ordered pair into each equation or inequality in the system.
Let's apply this to the given problem:

• For Pair (a) \(x = \frac{3}{2}\), \(y = \frac{4}{3}\):
  First inequality:
  \[2(\frac{3}{2}) + 3(\frac{4}{3}) \geq 2\]
  This simplifies to \[3 + 4 \geq 2\], which is true.
  Second inequality:
  \[4(\frac{3}{2}) - 6(\frac{4}{3}) < -1\]
  This simplifies to \[6 - 8 < -1\], which is false.
  Thus, Pair (a) is not a solution.
  
• For Pair (b) \(x = \frac{1}{4}\), \(y = \frac{7}{6}\):
  First inequality:
  \[2(\frac{1}{4}) + 3(\frac{7}{6}) \geq 2\]
  This simplifies to \[\frac{1}{2} + \frac{7}{2} \geq 2\], which is true.
  Second inequality:
  \[4(\frac{1}{4}) - 6(\frac{7}{6}) < -1\]
  This simplifies to \[1 - 7 < -1\], which is true.
  Thus, Pair (b) is a solution.

By following these steps, we can effectively use the substitution method to determine if an ordered pair is a solution to a system of inequalities.

Solving Inequalities

Solving inequalities involves finding the set of all possible values for the variables that make the inequality true.
This is similar to solving equations but with additional consideration of the inequality sign (\(>, <, \geq, \leq\)).
Here are the key steps:

• Isolate the variable on one side of the inequality.
  For example:
  \[2x + 3y \geq 2\]
  we can isolate 'y' to get
  \[3y \geq 2 - 2x\]
  \[y \geq \frac{2 - 2x}{3}\]
  
• Substitute the values from an ordered pair to check if the inequality holds true.
  This involves simple arithmetic operations.
  
• Compare the result to determine if the inequality is satisfied.
  Repeat this process for each inequality in the system.
  Combining these steps ensures a thorough way to find solutions to a system of inequalities.
  By understanding these concepts, solving inequalities becomes a more manageable task.