Question

Write a compound inequality and explain why it is a compound inequality.

Short answer

The compound inequality is \(13 \leq x \leq 19\) or \(x > 19\). It's a compound inequality because two inequalities are combined with 'or'.

Step-by-step solution

Understanding of compound inequalities

A compound inequality is a combination of two or more inequalities joined together by the words 'and' or 'or'. They are used to show that an expression can have more than one solution.

Formulating a Compound Inequality

Let's assume that 'x' represents the age of a person, and we want to describe that a person is either a teenager or an adult. A teenager is someone whose age is greater than or equal to 13 but less than or equal to 19. So we could write this as \(13 \leq x \leq 19\). The latter part ('or') is describing an adult, hence \(x > 19\). Combining these two gives us the compound inequality \(13 \leq x \leq 19\) or \(x > 19\).

Explanation

This is a compound inequality because it combines two inequalities: \(13 \leq x \leq 19\) and \(x > 19\). The 'or' connects the two inequalities making them a compound inequality. Hence, 'x' can be any number satisfying either \(13 \leq x \leq 19\) or any number greater than 19.

Key concepts

Inequalities

Inequalities are mathematical expressions that show the relationship between two values when they are not equal. Instead of an 'equals' sign, inequalities use symbols such as less than (<), greater than (>), less than or equal to (≤), and greater than or equal to (≥) to compare two quantities. For example, if we say that 'a' is less than 'b', we write it as \(a < b\).

Unlike equations that have one solution, inequalities usually have a range of possible solutions. Understanding inequalities is a fundamental part of algebra and helps in solving problems that involve limits, such as defining age ranges, speed limits, or temperature ranges.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables (like 'x' or 'y'), and operation signs (such as +, -, *, /). These expressions represent quantities symbolically and come in various forms, such as monomials, binomials, and polynomials. The beauty of algebraic expressions is that they can encapsulate patterns and general math relationships without specifying the values of the variables.

For example, \(2x + 3\) is an algebraic expression where '2x' is a term that shows how 'x' is multiplied by 2, and '3' is a term that stands on its own. Adding these expressions together allows algebra to describe a vast array of mathematical concepts and helps us solve problems involving unknown variables.

Solving Inequalities

Solving inequalities involves finding all the possible values of a variable that make the inequality true. The process is similar to solving equations but with a key difference: You must be careful when multiplying or dividing by a negative number, as this will reverse the direction of the inequality sign.

For example, to solve \(2x < 10\), you would divide both sides by 2 to find \(x < 5\), meaning that 'x' can be any number less than 5. When solving compound inequalities, such as \(3 < x + 2 < 7\), we treat each part of the compound inequality separately to find the range of possible values for 'x'.

Important Note

When dealing with 'and' compound inequalities, the solution is the intersection (overlap) of the solutions to each inequality. For 'or' compound inequalities, the solution is the union (combination) of the solutions, covering a broader range of possibilities.