Question

Write the verbal sentence as an equation or an inequality. Nine more than three fourths of \(y\) is less than six times \(w\)n.

Short answer

The verbal sentence 'Nine more than three fourths of \(y\) is less than six times \(w\).' translates into the inequality \(0.75y + 9 < 6w\).

Step-by-step solution

Represent first part of sentence in mathematical form

The first part of the sentence 'Nine more than three fourths of \(y\)' can be interpreted as: First, multiply \(y\) by three fourths (or 0.75). Then add nine to result. This results in the following expression: \(0.75y + 9\).

Represent second part of sentence in mathematical form

Next, the second part of the sentence 'is less than six times \(w\)' can be interpreted as: Multiply \(w\) by six. This results in the following expression: \(6w\). Since the original sentence stipulates that this value is greater, the inequality symbol used is '<'.

Combine into a single inequality

The final step is to combine these two mathematically represented parts joining by 'is less than'. Hence, the final inequality is \(0.75y + 9 < 6w\).

Key concepts

Inequalities in Algebra

When dealing with algebra, inequalities are equations that use signs such as '<', '>', '<=', or '>=' instead of the equal sign '='. They show that one side is not exactly equal to the other, but rather less than, greater than, or possibly equal when using the inclusive forms (<= or >=). In the context of the given exercise, understanding inequalities allows a student to express the relationship between quantities that don't have a fixed equivalence but have a defined order.

For instance, the exercise involves translating the statement 'Nine more than three fourths of y is less than six times w' into an algebraic inequality. Here, the terms 'more than' and 'less than' guide us to use the '<' sign to formulate the inequality correctly. Recognizing these keywords in verbal sentences and connecting them with their mathematical symbols is key to mastering inequalities in algebra.

Writing Algebraic Expressions

Converting words into algebraic expressions is a foundational skill in algebra. It involves recognizing numerical phrases and translating them using variables, coefficients, and arithmetic operations such as addition, subtraction, multiplication, and division.

Identifying Quantities and Operations

Consider the phrase 'Nine more than three fourths of y'. This phrase has two essential components: a multiplication part ('three fourths of y', which means 0.75 times y) and an addition part ('nine more than', which implies adding 9). Students should practice dissecting such terms to form accurate expressions. By writing algebraic expressions from words step by step, students bridge the gap between real-world scenarios and mathematical concepts, ultimately gaining fluency in the language of algebra.

Solving Algebraic Inequalities

Once you have an algebraic inequality, solving it involves finding the values of the unknown variables that make the inequality true. Similar to equations, solving inequalities requires manipulation of the inequality to isolate the variable of interest on one side. However, remember that if you multiply or divide both sides of an inequality by a negative number, the direction of the inequality flips.

Understanding Solution Sets

The solution to an inequality isn't just a single number but often a range of values, or 'solution set.' In our exercise, the inequality \(0.75y + 9 < 6w\) doesn't give a single solution but describes all the values of \(y\) and \(w\) that uphold this relationship. Developing the skill to solve inequalities is not only of great mathematical value but also highly applicable in real-life situations where ranges and limits are encountered, such as budgeting, measuring, and even decision-making.