Problem 43 - Beginning and Intermediate Algebra

Question

Is \((3,5)\) in the solution set of the compound inequality \(x-y \geq-6\) and \(2 x+y<7 ?\) Why or why not?

Step-by-Step Solution

Answer

The point \((3,5)\) is not in the solution set of the compound inequality because it satisfies the first inequality \(x-y \geq -6\) but not the second inequality \(2x+y < 7\). Both inequalities need to be true for the point to be in the solution set.

Step 1: Test the point in the inequality \(x-y \geq -6\)

Plug in the values \(x = 3\) and \(y = 5\) into the inequality \(x-y \geq -6\): \( (3) - (5) \geq -6\) \( -2 \geq -6\) The inequality holds true, as -2 is greater or equal to -6. Now, we will test the same point in the second inequality.

Step 2: Test the point in the inequality \(2x + y < 7\)

Plug in the values \(x=3\) and \(y=5\) into the inequality \(2x+y<7\): \(2(3) + (5) < 7\) \(6 + 5 < 7\) \(11 < 7\) The second inequality does not hold true, as 11 is not less than 7.

Step 3: Conclusion

Since the point \((3,5)\) makes the first inequality true but not the second one, it is not in the solution set of the compound inequality. Both inequalities must be true for the point to be in the solution set, so \((3,5)\) doesn't satisfy this requirement.

Key Concepts

Inequality Solutions

Inequalities are mathematical expressions that show the relationship between two values, indicating that one is greater than, less than, equal to, or not equal to the other. Solving inequalities is similar to solving equations, but with inequalities, we find a range of values rather than a specific number. This range of values represents the "solution set."

• A solution set is the collection of all values that make the inequality true.
• It's important to use proper inequalities symbols such as \(<, >, \geq, \leq\) to specify these relationships.

Compound inequalities involve more than one inequality joined by the terms "and" or "or." For example, in a compound inequality like \(x-y \geq -6\) and \(2x + y < 7\), we are looking for values of \(x\) and \(y\) that satisfy both conditions simultaneously. The "solution set" of a compound inequality consists only of those values that satisfy all parts of the inequality.

Testing Points in Inequalities

Testing points in inequalities is a method used to determine whether a specific point lies within the solution set. This involves substituting the coordinate values into each inequality and checking if the resulting statement is true.For a compound inequality, you must test the point in each of the individual inequalities:

• Substitute the point into the first inequality, \(x - y \geq -6\), to see if it holds true.
• Then, substitute the point into the second inequality, \(2x + y < 7\), and check if this condition also holds.

If both conditions are true, the point is part of the solution set. If one or both conditions are false, the point is not part of the solution set.
This process is simple yet essential for validating potential solutions.

Algebraic Problem-Solving

Algebraic problem-solving skills are crucial when working with inequalities, as they help in rearranging and simplifying expressions to find solutions more easily. Using these skills involves:

• Substituting known values of variables back into the inequalities.
• Simplifying complex expressions to make inequalities easier to solve.
• Understanding how the manipulation of equations affects inequalities, such as changing a sign when multiplying or dividing by a negative number.

This approach allows for a structured method, aiding consistency and accuracy when determining the relationships expressed through inequalities. Having a strong grasp on algebraic skills ensures you can tackle even the most challenging inequalities efficiently.

Solution Sets

A solution set in the context of compound inequalities refers to the set of values that satisfy all parts of the given inequalities. It is the set of points that make each individual inequality true when applied in the compound expression.

• If a point satisfies both inequalities, it lies within the solution set.
• If it doesn't satisfy even one inequality, it is not included in the solution set.

In the example problem, the point \(3, 5\) only satisfies the first inequality, \(x - y \geq -6\), but does not satisfy the second inequality, \(2x + y < 7\). Therefore, it is not part of the compound inequality's solution set because only points that hold true in both inequalities belong in the solution set. A good understanding of this concept is imperative for accurately solving and graphing inequalities.