Problem 115 - Elementary and Intermediate Algebra

Question

Suppose \(x\) is positive and \(y\) is negative. Determine whether each statement is true or false. a. \(x-y>0\) b. \(y-x<0\) c. \(|-x|<0\) d. \(-|y|<0\)

Step-by-Step Solution

Answer

a. True, b. True, c. False, d. True.

Step 1: Understanding the condition for Statement a

To evaluate the statement \(x-y>0\), start by recognizing that \(x\) is positive and \(y\) is negative. This means that when you subtract a negative number (\(y\)) from a positive number (\(x\)), you are essentially adding the absolute value of \(y\) to \(x\). Mathematically, this means \(x-y = x + |y|\). Since both \(x\) and \(|y|\) are positive, \(x-y\) must also be positive.

Step 2: Evaluating Statement a

Since \(x-y = x + |y|\) and both \(x\) and \(|y|\) are positive numbers, the statement \(x-y > 0\) is true. Subtracting a negative number increases the value of \(x\).

Step 3: Understanding the condition for Statement b

For \(y-x < 0\), we need to consider the expression \(y-x = y - x\). Here, \(y\) is negative, and subtracting \(x\) means reducing it further by a positive number. Thus, \(y-x\) is even more negative than \(y\) alone.

Step 4: Evaluating Statement b

Given that \(y\) is negative and \(x\) is positive, \(y-x\) results in a negative number even further decreased by subtracting \(x\). Therefore, \(y-x < 0\), making the statement true.

Step 5: Understanding the condition for Statement c

Examine the expression \(|-x| < 0\). The absolute value function, \(|\ldots|\), always returns non-negative results. For \(-x\), even though \(-x\) itself is negative, \(|-x| = x\), which is positive.

Step 6: Evaluating Statement c

Since \(|-x| = x\) and \(x > 0\), it is not possible for \(|-x|\) to be less than 0. Hence, the statement \(|-x|<0\) is false.

Step 7: Understanding the condition for Statement d

Consider the expression \(-|y| < 0\). \(|y|\) gives the positive value of the negative number \(y\), and \(-|y|\) reflects it across the origin, making it negative.

Step 8: Evaluating Statement d

Because \(-|y|\) is negative (since it's the negative of a positive number \(|y|\)), the statement \(-|y| < 0\) is true.

Key Concepts

Absolute Value

The absolute value of a number is like the number's distance from zero on the number line. It doesn't matter if the number is positive or negative; the absolute value is always non-negative.

• Understanding Absolute Value: The absolute value of a number changes any negative sign to positive. For instance, \(|-5| = 5\).
• Application: Regardless of the original value being positive or negative, absolute values are crucial when comparing magnitudes or distances without considering direction.

In exercises like the one above, the concept of absolute value helps determine expressions, such as what happens when you take the absolute value of a negative variable, converting it to positive, like \(|y| \) when \(y\) is negative.

Positive and Negative Numbers

Positive and negative numbers are foundation elements in math and depict direction and quantity.

• Positive Numbers: These are greater than zero and usually do not have a sign, such as 3, 17, or 42. They appear to the right of zero on the number line.
• Negative Numbers: These are less than zero and are often marked with a minus sign, such as -2, -10, or -100. They are located on the left side of zero on the number line.

In expressions involving negative and positive numbers, operations can alter results significantly. Subtracting a negative is equivalent to adding its absolute value, making the outcome potentially larger. For instance, if \(y\) is negative and \(x\) is positive, \(x-y\) effectively becomes \(x + |y|\), showing how subtracting a negative leads to an increased result.

Inequalities

Inequalities are statements showing that two expressions are not equal. They often involve relational symbols such as <, >, ≤, and ≥. Understanding how these work helps in solving comparison-related problems in algebra.

• Understanding the Symbols: "<" means "less than", "<=" means "less than or equal to", ">" means "greater than", and ">=" signifies "greater than or equal to".
• Solving Inequalities: Often involves manipulating both sides of the inequality just like in regular algebraic equations; however, special rules apply, such as reversing the sign of the inequality when multiplying or dividing by a negative number.

In the step-by-step solutions, inequalities help assess whether statements are true or false. For instance, if you know \(x\) is positive and \(y\) is negative, then \(x-y>0\) fits because you are adding a positive absolute value to a positive number.