Problem 37 - Algebra for College Students

Question

Determine which of the ordered pairs \((1,3),(-2,5)\) \((-6,-4),\) and \((7,-8)\) satisfy each compound or absolute value inequality. $$y>5 x \text { and } y<-x$$

Step-by-Step Solution

Answer

The ordered pair (-6, -4) satisfies both inequalities.

Step 1 - Analyze the inequality conditions

There are two conditions to check for each ordered pair: (i) the y-coordinate must be greater than 5 times the x-coordinate (i.e., y > 5x), and (ii) the y-coordinate must also be less than the negative of the x-coordinate (i.e., y < -x). An ordered pair must satisfy both conditions simultaneously to be a solution.

Step 2 - Test the first pair \(1, 3\)

Check (1, 3):1. For y > 5x: 3 > 5(1) => 3 > 5 (False).2. For y < -x: 3 < -1 => 3 < -1 (False).Since neither condition is satisfied, (1, 3) is not a solution.

Step 3 - Test the second pair \(-2, 5\)

Check (-2, 5):1. For y > 5x: 5 > 5(-2) => 5 > -10 (True).2. For y < -x: 5 < -(-2) => 5 < 2 (False).Since only one condition is satisfied, (-2, 5) is not a solution.

Step 4 - Test the third pair \(-6, -4\)

Check (-6, -4):1. For y > 5x: -4 > 5(-6) => -4 > -30 (True).2. For y < -x: -4 < -(-6) => -4 < 6 (True).Since both conditions are satisfied, (-6, -4) is a solution.

Step 5 - Test the fourth pair \(7, -8\)

Check (7, -8):1. For y > 5x: -8 > 5(7) => -8 > 35 (False).2. For y < -x: -8 < -7 => -8 < -7 (True).Since only one condition is satisfied, (7, -8) is not a solution.

Key Concepts

Understanding Coordinate Pairs

Coordinate pairs are used to represent points in the Cartesian coordinate system. These pairs are written as \( (x, y) \) where the first value corresponds to the x-coordinate and the second value corresponds to the y-coordinate. To determine if a coordinate pair satisfies a particular set of conditions, you need to substitute the x and y values from the pair into the conditions and check if the resulting statements are true.

Analyzing Inequality Conditions

In mathematics, an inequality compares two values and shows that one value is greater than, less than, or equal to the other. In the given exercise, there are two inequalities: \( y > 5x \) and \( y < -x \). To determine whether an ordered pair \( (x, y) \) satisfies these compound inequalities, you must check both conditions:

• First, check if the y-coordinate is greater than 5 times the x-coordinate (\( y > 5x \)). Substitute the x and y values and solve the inequality.
• Second, check if the y-coordinate is also less than the negative of the x-coordinate (\( y < -x \)). Again, substitute the values and solve the inequality.

An ordered pair must satisfy both conditions simultaneously to be considered a solution to the given compound inequalities.

Absolute Value Inequalities Explained

Absolute value inequalities are a bit different from regular inequalities. The absolute value of a number is its distance from zero on the number line, regardless of direction. For example, the absolute value of -3 is 3, written as \( | -3 | = 3 \).

When working with absolute value inequalities, there are a few key things to keep in mind:

• An inequality of the form \( |x| < a \) means that the value of x is less than a units away from zero, which translates to \( -a < x < a \).
• Conversely, an inequality of the form \( |x| > a \) implies that the value of x is more than a units away from zero, so \( x < -a \) or \( x > a \).

Using these principles, you can solve absolute value inequalities by breaking them down into two separate inequalities.