Problem 33 - Algebra and Trigonometry

Question

\(33-34=\) 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: Inequality for Positive Numbers

To write the statement \(x\) is positive in terms of an inequality, we note that positive numbers are greater than 0. Therefore, the inequality is: \(x > 0\).

Step 2: Inequality for Less Than

The statement \(t\) is less than 4 translates to the inequality: \(t < 4\). This means that \(t\) can be any value smaller than 4.

Step 3: Inequality for Greater Than or Equal To

The statement \(a\) is greater than or equal to \(\pi\) can be written as: \(a \geq \pi\). This indicates that \(a\) can be equal to \(\pi\) or any number greater than \(\pi\).

Step 4: Compound Inequality

The statement \(x\) is less than \(\frac{1}{3}\) and is greater than \(-5\) translates to a compound inequality: \(-5 < x < \frac{1}{3}\). This implies \(x\) must be between \(-5\) and \(\frac{1}{3}\), but not including \(-5\) or \(\frac{1}{3}\).

Step 5: Inequality for Distance

The distance from \(p\) to 3 is at most 5 translates to an absolute value inequality: \(|p - 3| \leq 5\). This means that the distance between \(p\) and 3 is no greater than 5, covering the range from \(3 - 5\) to \(3 + 5\).

Key Concepts

Positive Numbers

In the world of mathematics, a number is considered positive if it is greater than 0. This concept is crucial when writing inequalities. For instance, if a statement says "\(x\) is positive," you can express this using the inequality \(x > 0\).

This inequality indicates that \(x\) can be any number greater than zero, meaning it can include numbers like 1, 2.5, and even very large numbers like 1000. Negative numbers and zero do not belong to this category.

Understanding this simple principle helps in solving many types of mathematical problems, particularly when working with positive constraints in equations.

Compound Inequalities

Compound inequalities involve combining two or more inequalities into a single statement. These are often used to describe a range of possible values that a variable can take.

For example, if you have a statement like "\(x\) is less than \(\frac{1}{3}\) and greater than \(-5\)," you can write this as the compound inequality \(-5 < x < \frac{1}{3}\).

This compound inequality tells us that \(x\) is any number between \(-5\) and \(\frac{1}{3}\), not including the endpoints. It's important to solve each part of the compound inequality separately before combining results. This understanding is essential for solving complex problems where variables can have multiple conditions to satisfy.

Absolute Value Inequalities

Absolute value inequalities are used when you need to measure the distance from zero on a number line without considering direction.

An interesting scenario is when you need to express that the distance from a point, such as \(p\) to 3, is at most 5. This is expressed as the inequality \(|p - 3| \leq 5\).

- The symbol \(\leq\) indicates that the distance is no more than 5. - This inequality translates to the range \(-5 \leq p - 3 \leq 5\), which can be rewritten by solving for \(p\), resulting in \(-2 \leq p \leq 8\). Absolute value inequalities are powerful tools for defining constraints involving distance and tolerance in real-world scenarios.

Greater Than or Equal to

The expression \(\geq\) in inequalities means "greater than or equal to." This concept allows variables to take on a value equal to, or any value greater than, a certain number.

For instance, if the statement is "\(a\) is greater than or equal to \(\pi\)," you can write it as \(a \geq \pi\).

In this scenario, \(a\) can equal \(\pi\) itself, allowing it to be exactly the constant 3.14159..., or it can be any larger number. This type of inequality is particularly useful in real-world applications like minimum product requirements or allowable error margins in calculations.