Problem 45 - Algebra 1

Question

Translate the sentence into an equation or an inequality. 17 less than a number \(z\) is \(9 .\)

Step-by-Step Solution

Answer

The translation of the sentence into an equation is \(z - 17 = 9\).

Step 1: Interpret the Given Sentence

When the problem says '17 less than a number \(z\)', it suggests subtracting \(17\) from an unknown number \(z\).

Step 2: Form the equation

Therefore, the phrase '17 less than a number \(z\)' can be written as \(z - 17\).

Step 3: Translate 'is 9'

The phrase 'is 9' is instructing that the value of the mathematical expression (which is \(z - 17\)) equals \(9\). This can be written as \(z - 17 = 9\).

Key Concepts

Inequalities

In mathematics, an inequality is a relation that holds between two values when they are different. It shows if one value is less than, greater than, or simply not equal to another value. When translating words to inequalities, we often encounter phrases like 'less than', 'greater than', 'at least', and 'no more than'. Unlike equations that show equivalence, inequalities express a range of possible solutions.

For example, 'a number x is greater than 12' can be written as the inequality x > 12. Understanding inequalities is crucial as they are widely used to describe conditions in subjects such as economics, engineering, and the sciences. To solve inequalities, similar steps to equation solving are taken by performing operations on both sides until the unknown is isolated.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. For instance, the expression 3x + 2 represents a quantity that depends on the value of x. Translating words into algebraic expressions is a fundamental skill in algebra that involves recognizing key phrases that indicate specific operations.

Phrases such as 'the sum of' or 'increased by' suggest addition, while 'the product of' indicates multiplication. Our textbook exercise brought out the phrase '17 less than a number z', which was translated into the expression z - 17. Mastery in converting verbal descriptions into algebraic expressions is not only important in solving math problems but also enhances logical thinking.

Equation Solving

Moving to equation solving, it often begins with an algebraic expression that has been set equal to a value—essentially, an equation. To solve an equation like z - 17 = 9, we aim to find the value of the variable that makes the equation true. This involves performing operations to isolate the variable on one side of the equation.

In our example, we would add 17 to both sides of the equation to solve for z, giving us z = 9 + 17, which simplifies to z = 26. Equation solving can range from these simple, one-step problems to much more complex ones involving multiple steps and operations. It is a key part of algebra that is used in countless real-world applications, such as in science, finance, and engineering.