Problem 34 - Algebra and Trigonometry

Question

\(33-34=\) Write each statement in terms of inequalities. (a) \(y\) is negative (b) \(z\) is greater than 1 (c) \(b\) is at most 8 (d) \(w\) is positive and is less than or equal to 17 (e) \(y\) is at least 2 units from \(\pi\)

Step-by-Step Solution

Answer

(a) \( y < 0 \), (b) \( z > 1 \), (c) \( b \leq 8 \), (d) \( 0 < w \leq 17 \), (e) \( y \leq \pi - 2 \) or \( y \geq \pi + 2 \)

Step 1: Express 'y is negative' as an inequality

If a number is negative, it is less than 0. Therefore, the inequality is: \( y < 0 \)

Step 2: Express 'z is greater than 1' as an inequality

If a number is greater than a given number, it is simply 'greater than' that number. Therefore, the inequality is:\( z > 1 \)

Step 3: Express 'b is at most 8' as an inequality

If a number is at most a specific value, it means the number can either be equal to or less than that value. Therefore, the inequality is:\( b \leq 8 \)

Step 4: Express 'w is positive and is less than or equal to 17' as an inequality

A number is positive if it is greater than 0. Saying it is less than or equal to another number, in this case 17, combines into:\( 0 < w \leq 17 \)

Step 5: Express 'y is at least 2 units from π' as an inequality

If a number is at least 2 units away from a given number, it means it can be 2 units more or 2 units less than that number. Therefore, the inequalities are:\( y \leq \pi - 2 \) or \( y \geq \pi + 2 \)

Key Concepts

Negative Numbers

Negative numbers are numbers that are less than zero. They are found on the left side of the zero on a number line. Understanding negative numbers is crucial as they represent values below zero, such as temperatures below freezing or elevations below sea level.

Here are some properties of negative numbers:

• They are denoted with a minus sign (e.g., -5).
• The further left a negative number is on the number line, the smaller its value. For instance, -10 is less than -5.
• When comparing negative numbers, -3 is greater than -5, although it might seem the opposite because -3 is closer to zero.

In terms of inequalities, if a number is negative, you can express this as: \( y < 0 \). This tells us that \( y \) can be any number below zero.

Greater Than Inequalities

In mathematics, an inequality is a relationship between two values, indicating that one is larger than the other. The symbol \( > \) denotes 'greater than'. For instance, when we say \( z > 1 \), we mean that \( z \) can be any number larger than 1.

Here are some key points about greater than inequalities:

• The solution set includes all numbers to the right of a given number on the number line.
• It does not include the number itself. For example, \( z > 1 \) doesn't include 1.
• This type of inequality is often used when describing conditions that require a minimum threshold, like heights or scores.

Understanding these principles helps in solving real-world problems where a minimum criteria or limit is set.

Less Than or Equal To

The 'less than or equal to' symbol is \( \leq \). This inequality means the value on the left is either less than the right value or equal to it. For example, \( b \leq 8 \) implies \( b \) can be 8 or any number less than 8.

Notable characteristics:

• The number itself is included in the possible solutions.
• You can display this inequality on a number line using a closed dot to show the endpoint is included.
• It is regularly used in contexts where a maximum limit is specified, like budget constraints or allowable speeds.

Such conditions are common in everyday situations, indicating a cap or boundary that should not be exceeded.

Absolute Value Inequalities

Absolute value inequalities involve the distance from zero a number is on the number line. The absolute value of \( x \) is expressed as \( |x| \), which counts how far a number is from zero without considering direction.

When an absolute value inequality states something like "at least 2 units from \( \pi \)," it indicates that the number could be either 2 units more or 2 units less than \( \pi \). The expression becomes two separate inequalities:

• \( y \leq \pi - 2 \)
• \( y \geq \pi + 2 \)

These inequalities capture all the numbers that are "at least 2 units away" from \( \pi \). This concept is useful in scenarios where boundaries or thresholds are set relative to a specific point or value.