Problem 40 - 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. $$y>-x+1 \text { or } y<4 x$$

Step-by-Step Solution

Answer

(1, 3), (-2, 5), (7, -8) satisfy the compound inequality; (-6, -4) does not.

Step 1 - Understand the Inequalities

The compound inequality consists of two separate inequalities connected by 'or': 1. \( y > -x + 1 \)2. \( y < 4x \)

Step 2 - Test Each Ordered Pair for Each Inequality

We will test \( (1, 3) \), \( (-2, 5) \), \( (-6, -4) \), and \( (7, -8) \) to see if they satisfy either inequality.

Step 3 - Test the Pair \( (1, 3) \)

Substitute x = 1 and y = 3:1. For \ y > -x + 1: \( 3 > -1 + 1 \Rightarrow 3 > 0 \) (True) 2. For \ y < 4x: \( 3 < 4(1) \Rightarrow 3 < 4 \) (True) Since both inequalities are true, \( (1, 3) \) satisfies the compound inequality.

Step 4 - Test the Pair \( (-2, 5) \)

Substitute x = -2 and y = 5:1. For \ y > -x + 1: \( 5 > 2 + 1 \Rightarrow 5 > 3 \) (True)2. For \ y < 4x: \( 5 < 4(-2) \Rightarrow 5 < -8 \) (False)Since the first inequality is true, \( (-2, 5) \) satisfies the compound inequality.

Step 5 - Test the Pair \( (-6, -4) \)

Substitute x = -6 and y = -4:1. For \ y > -x + 1: \( -4 > 6 + 1 \Rightarrow -4 > 7 \) (False)2. For \ y < 4x: \( -4 < 4(-6) \Rightarrow -4 < -24 \) (False)Since both inequalities are false, \( (-6, -4) \) does not satisfy the compound inequality.

Step 6 - Test the Pair \( (7, -8) \)

Substitute x = 7 and y = -8:1. For \ y > -x + 1: \( -8 > -7 + 1 \Rightarrow -8 > -6 \) (False)2. For \ y < 4x: \( -8 < 4(7) \Rightarrow -8 < 28 \) (True)Since the second inequality is true, \( (7, -8) \) satisfies the compound inequality.

Key Concepts

Inequalities

Inequalities are mathematical expressions involving the symbols <, >, ≤, or ≥. They are used to show the relations between different values. For instance, the inequality 3 < 5 means that 3 is less than 5. In our exercise, we have two linear inequalities: \(y > -x + 1\) and \(y < 4x\). The '>' symbol means 'greater than,' while the '<' symbol means 'less than.' These inequalities form what's known as a compound inequality because they are combined using the word 'or.' This implies that a solution can satisfy either one of the inequalities or both.

Ordered Pairs

Ordered pairs are pairs of numbers typically written in parentheses like this: (x, y). Each pair represents a point on a coordinate plane, where 'x' is the horizontal value and 'y' is the vertical value. For example, the ordered pair (1, 3) indicates a point where x = 1 and y = 3. In our exercise, we test different ordered pairs—(1, 3), (-2, 5), (-6, -4), and (7, -8)—to see if they satisfy the inequalities. To do this, we substitute the x and y values from each pair into the inequalities and check if the resulting statements are true or false.

By following steps, we derive if a pair satisfies either one of the inequalities in our compound inequality.

Absolute Value Inequalities

Absolute value inequalities involve the absolute value function, denoted by vertical bars like so: \(|x|\). This function returns the non-negative value of a number, regardless of its sign. For instance, \(|-3| = 3\) and \(|5| = 5\). An example of an absolute value inequality would be \(|x| < 4\), which translates to \(-4 < x < 4\). It means that x can be any value between -4 and 4.

In the given exercise, although we do not deal with absolute value inequalities directly, knowing how to handle them is crucial. Steps to solve absolute value inequalities often involve splitting them into two separate inequalities, based on the definition of absolute values. For example, solving \(|x - 2| < 5\) would mean solving \(-5 < x - 2 < 5\).