Problem 60 - Algebra 1

Question

EQUATIONS AND INEQUALITIES Match the verbal sentence with its mathematical representation. The product of 16 and \(x\) is less than 32

Step-by-Step Solution

Answer

The mathematical representation of 'The product of 16 and \(x\) is less than 32' is \(16x < 32\).

Step 1: Understand the verbal sentence

Here, 'The product of 16 and \(x\)' means 16 is being multiplied by \(x\). 'Is less than' means the output of the multiplication should be less than 32.

Step 2: Represent the verbal sentence mathematically

By substituting the verbal sentence into a mathematical representation, we have: \(16x < 32\) as the mathematical representation.

Key Concepts

Solving Inequalities

Understanding how to solve inequalities is a fundamental skills in algebra that allows students to find the range of values that hold true for a given algebraic statement. Inequalities differ from equations in that they express a less than or greater than relationship, rather than equality.

For example, the inequality transformation of the verbal statement 'The product of 16 and x is less than 32' results in the mathematical form: \(16x < 32\). Solving this involves similar steps to solving an equation: isolating the variable on one side to find its possible values. To solve \(16x < 32\), one would divide both sides by 16 to get \(x < 2\). This result means that any real number less than 2 satisfies the inequality.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables (like \(x\)), and operation symbols. They are imperative in translating real-world problems into mathematical language. An expression doesn't have an equality sign as an equation does.

For instance, in the context of our example, 'The product of 16 and x' is translated into the algebraic expression \(16x\). When working to understand algebraic expressions, focus on the operations suggested by the terms such as 'product' which implies multiplication. It's crucial for students to identify key terms and associate them with their mathematical counterparts in order to successfully interpret and construct algebraic expressions.

Verbal Sentences in Math

Translating verbal sentences into mathematical expressions or inequalities is a vital skill in solving algebra problems. A verbal sentence in math is a statement that describes a mathematical relationship in words.

To adeptly transform a verbal sentence into its mathematical representation, one must prioritize understanding the meaning of terms and phrases used in typical math contexts. In our original exercise, the phrase 'the product of 16 and x is less than 32' contains specific mathematical language, where 'product' indicates a multiplication operation and 'less than' indicates an inequality. Recognizing and translating these terms allows students to convert the sentence into the inequality \(16x < 32\), effectively bridging the gap between verbal language and mathematical symbolism.