Problem 37 - Precalculus Mathematics for Calculus

Question

Write each statement in terms of inequalities. (a) \(x\) is positive (b) \(t\) is less than 4 (c) \(a\) is greater than or equal to \(\pi\) (d) \(x\) is less than \(\frac{1}{3}\) and is greater than \(-5\) (e) The distance from \(p\) to 3 is at most 5

Step-by-Step Solution

Answer

(a) \(x > 0\), (b) \(t < 4\), (c) \(a \geq \pi\), (d) \(-5 < x < \frac{1}{3}\), (e) \(|p - 3| \leq 5\).

Step 1: Identifying Inequality for Positive Number

For part (a), we are given that \(x\) is positive. A positive number is any number that is greater than 0. Therefore, the inequality statement for \(x\) being positive is \(x > 0\).

Step 2: Identifying Inequality for Less Than

For part (b), the statement is that \(t\) is less than 4. This means that the value of \(t\) is any number that is smaller than 4. The inequality that represents this condition is \(t < 4\).

Step 3: Identifying Inequality for Greater Than or Equal

In part (c), the statement is that \(a\) is greater than or equal to \(\pi\). This means \(a\) can be either equal to \(\pi\) or any number greater than \(\pi\). The inequality for this statement is \(a \geq \pi\).

Step 4: Identifying Compound Inequality

For part (d), \(x\) is less than \(\frac{1}{3}\) and greater than \(-5\). This is a compound inequality involving two conditions: \(x\) is greater than \(-5\) and simultaneously less than \(\frac{1}{3}\). This can be written as \(-5 < x < \frac{1}{3}\).

Step 5: Expressing Distance with Inequality

The statement in part (e) deals with distance: the distance from \(p\) to 3 is at most 5. Distance can be expressed with absolute values, so we write this as \(|p - 3| \leq 5\), meaning \(p - 3\) falls within the range from -5 to 5.

Key Concepts

Positive Numbers

In mathematics, positive numbers are those numbers that are greater than zero. They are located to the right of zero on the number line, representing quantities more than nothing. When we talk about a positive number in terms of inequality, we use the symbol '>'. For example, if we say that a number \(x\) is positive, we can express this statement as the inequality \(x > 0\). This indicates that \(x\) can be any number as long as it is greater than zero.

• Examples of positive numbers are 1, 5, and 23.
• Positive numbers do not include zero or any negative numbers.
• They are often simply referred to as 'greater than zero'.

Compound Inequalities

Compound inequalities contain two or more inequalities joined by 'and' or 'or'. These compound statements define a range of values that a variable can take. For example, if we say a variable \(x\) is greater than \(-5\) and less than \(\frac{1}{3}\), we write this as a compound inequality: \(-5 < x < \frac{1}{3}\). This statement means that \(x\) must satisfy both conditions simultaneously.

• With 'and', both conditions must be true: \(a < x < b\).
• With 'or', at least one condition must be true: \(x < a \; \text{or} \; x > b\).
• Compound inequalities can show intervals on a number line, such as open or closed intervals based on whether endpoints are included.

Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. It is a measure of magnitude without considering sign, therefore, an absolute value is always non-negative. This concept is represented by two vertical bars, for example, the absolute value of \(p - 3\) is written as \(|p - 3|\).

• The absolute value of 5 is \(5\).
• The absolute value of \(-5\) is also \(5\).
• It solves equations involving distance: \(|x| = a\) implies that \(x = a\) or \(x = -a\).
• It can be used to define inequalities regarding distance, such as \(|x| < a\).

Distance Inequalities

Distance inequalities use the concept of absolute value to express the separation between numbers on a number line. They help define how far apart two values are, either exactly or within a range. For example, if the distance between \(p\) and 3 is no more than 5 units, it is denoted as \(|p - 3| \leq 5\). This means that the difference \(p - 3\) can be between \(-5\) and \(5\).

• Used to find ranges: if \(|x - 5| < 3\), then distance is less than 3.
• Helps graphically on a number line to show reachable points.
• Distance concept helps solve real-world problems such as measuring tolerances and limits.