Question

Write the verbal sentence as an equation, or an inequality. Twenty-three less than the difference of thirty-eight and a number \(n\) is less than eight.

Short answer

The inequality in its simplest form, as per the given verbal sentence, is \(15 - n < 8\).

Step-by-step solution

Identifying the Key Parts of the Sentence

Identify the number \(n\), which is the variable in the problem. See how the figure 'twenty-three' and the 'difference of thirty-eight and a number \(n\)' function in relation to this variable. The hint is in the last part of the sentence, which says 'is less than eight'.

Forming the Inequality

The description 'Twenty-three less than the difference of thirty-eight and a number \(n\)' can be translated into mathematical form as \((38 - n) - 23\). This expression should then be 'less than eight', resulting in the inequality \((38 - n) - 23 < 8\).

Simplifying the Inequality

Simplify the inequality within brackets, which becomes \(15 - n < 8\).

Key concepts

Forming Inequalities

When translating verbal statements into mathematical inequalities, it's crucial to understand the language used. Inequalities express a relationship between two expressions that are not necessarily equal but have a defined order, such as less than, greater than, less than or equal to, or greater than or equal to. In our example, the phrase 'twenty-three less than the difference of thirty-eight and a number n' is interpreted as taking 23 away from the difference (38 - n). Recognizing words like 'less than', 'more than', 'at least', and 'no more than' helps identify the kind of inequality sign needed.

The complete verbal statement is 'Twenty-three less than the difference of thirty-eight and a number n is less than eight,' which leads us to form the inequality \( (38 - n) - 23 < 8 \). This is the mathematical representation of the statement. It's essential to keep the order and direction of the inequality sign in line with the verbal description to ensure the inequality corresponds accurately to what is described verbally.

Simplifying Inequalities

Simplifying an inequality is akin to simplifying an equation. The goal is to isolate the variable on one side, which gives us a clearer picture of the solution set. The key is to perform the same operation on both sides of the inequality without changing its direction, unless we multiply or divide by a negative number, which would flip the inequality.

In our step by step solution, the inequality starts as \( (38 - n) - 23 < 8 \). To simplify it, we combine like terms inside the brackets, resulting in \(15 - n < 8\). Notice we haven't changed the direction or the '<' sign because we haven't multiplied or divided by a negative number. Simplicity aids in understanding the range of possible values for n.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They are fundamental in translating verbal sentences into the language of algebra. The absence of an equality or inequality sign in an expression means it represents a value that can change, depending on the variable's value.

In our example, 'the difference of thirty-eight and a number n' is an algebraic expression representing the subtraction of n from 38, shown as \(38 - n\). These expressions are the building blocks of equations and inequalities. Understanding how to manipulate these expressions is key to solving algebraic problems. A firm grasp on combining like terms, applying the distributive property, and factoring will bolster your algebraic proficiency substantially.