Problem 44 - Beginning and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The point (3,5) is in the solution set of the compound inequality \(x-y \geq -6\) or \(2x+y<7\) because it satisfies the first inequality, \(x-y \geq -6\), with \(-2 \geq -6\). It doesn't satisfy the second inequality, \(2x+y<7\), as \(11<7\) is not true. However, since it's an "or" compound inequality, satisfying at least one inequality is sufficient for the point to be part of the solution set.

Step 1: Identify the coordinates of the point

The given point is (3,5), so the x-coordinate is 3 and the y-coordinate is 5. We will use these values to check if the point satisfies any of the given inequalities.

Step 2: Check if the point satisfies the first inequality

To check if (3,5) satisfies the inequality \(x-y \geq -6\), we substitute the x-coordinate with 3 and the y-coordinate with 5: \(3 - 5 \geq -6\) \(-2 \geq -6\) Since -2 is greater than or equal to -6, the point (3,5) satisfies the first inequality.

Step 3: Check if the point satisfies the second inequality

To check if (3,5) satisfies the inequality \(2x+y<7\), we substitute the x-coordinate with 3 and the y-coordinate with 5: \(2(3) + 5 < 7\) \(6 + 5 < 7\) \(11 < 7\) Since 11 is not less than 7, the point (3,5) does not satisfy the second inequality.

Step 4: Determine if the point is in the solution set

The compound inequality is an "or" inequality, meaning the point (3,5) will be in the solution set if it satisfies at least one of the inequalities. Since it satisfies the first inequality, (3,5) is in the solution set of the compound inequality.

Key Concepts

Understanding the Inequality Solution Set

In mathematics, the "solution set" of an inequality refers to the set of all possible values that satisfy that inequality. For compound inequalities, which consist of two or more inequalities joined by "and" or "or," the solution set can be slightly more complex.

An "or" compound inequality, like in the problem at hand, implies that if a pair of values satisfies even one of the inequalities, they belong to the solution set. It's as if each inequality provides a separate gate to the solution house — passing through any gate gets you in.

In our problem, (3,5) was checked against two inequalities: \(x-y \geq -6\) and \(2x+y<7\). Because it satisfied \(x-y \geq -6\), it was included in the solution set despite not meeting the second criterion. Understanding this concept is crucial for solving compound inequalities effectively.

Explaining Coordinate Substitution

Coordinate substitution is a method used to determine if a specific point lies within the solution set of a given inequality. This entails replacing the variables with actual numbers based on the coordinates of the point in question.

For example, given the point (3,5), you substitute \(x\) with 3 and \(y\) with 5 in the inequalities. So, for checking the first inequality \(x-y \geq -6\), we performed the substitution: \(3 - 5\).

• This allows us to verify the validity of the inequality under specific conditions.
• It's a practical approach to "plug in and test" without altering the inequality's general form.

Through coordinate substitution, you can systematically check whether a given point meets the inequality conditions one at a time, facilitating a step-by-step assessment.

The Process of Inequality Verification

Inequality verification involves substituting values and calculating numerical expressions to confirm or disprove the satisfaction of given inequalities. This process is essential in math to ensure solutions are correct and viable under defined numerical boundaries.

In inequality verification, like with \(x-y \geq -6\), the substituted expression is calculated to verify whether the resultant numerical statement is true or false:

• By computing \(3 - 5\), we got \(-2\) and checked it against \(-6\).
• If the statement holds true, as in \(-2 \geq -6\), the point satisfies the inequality.

Verifying inequalities is about ensuring each step logically confirms the mathematical statement under review. It's like a mini-proof, confirming or debunking the initial hypothesis based on calculated evidence.