Problem 75 - Introductory and Intermediate Algebra for College Students 4th

Question

Write each English phrase as an algebraic expression. Then simplify the expression. Let x represent the number. eight increased by the product of 5 and one less than a number

Step-by-Step Solution

Answer

The simplified expression of the given English phrase is \(5x + 3\)

Step 1: Translating the English phrase into an algebraic expression

The English phrase given is: 'eight increased by the product of 5 and one less than a number'. Breaking it down, 'eight increased by' translates to \(8 + \), 'the product of 5 and' translates to \(5 \times \), and 'one less than a number' translates to \(x - 1\). So, combining it all, it translates into the algebraic expression \(8 + (5 \times (x - 1))\) .

Step 2: Simplifying the algebraic expression

To simplify the expression, first handle the parentheses by using the distributive property of multiplication over subtraction. In other words, multiply 5 with both \(x\) and \(-1\) giving us the expression, \(8 + 5x - 5\). Now, combine like terms by adding up the numbers to give the algebraic expression : \(5x + 3\).

Key Concepts

Algebraic Expressions

When it comes to translating English phrases into algebraic expressions, it's a bit like turning words into numbers and symbols that a mathematician can work with. An algebraic expression is a combination of numbers, variables (like x, y, or z), and operation symbols (such as +, -, *, /). Each expression represents a value and it can vary, depending on the values of its variables.

For instance, when we take a phrase like 'eight increased by the product of 5 and one less than a number,' we recognize 'increased by' as the addition operation, 'product of' as multiplication, and 'one less than' as an indication to subtract one from our unknown number, which we've called x.

The resulting algebraic expression for this phrase is \(8 + 5 \times (x - 1)\), which neatly encapsulates the logic of the original English phrase in mathematical language.

Simplifying Expressions

Simplifying an algebraic expression means to rewrite it in the most condensed form possible without changing its value. It involves combining like terms and using mathematical properties like the distributive property to reduce an expression to fewer terms or make it easier to understand.

To simplify the expression \(8 + 5 \times (x - 1)\), we first distribute the 5 to both terms within the parentheses: \(x\) and \(–1\). This application of the distributive property yields \(5x - 5\). Following this, we combine this with our constant, 8, getting \(8 - 5 + 5x\). Then, it's just a matter of adding and subtracting those constants to get our final, simplified expression: \(5x + 3\).

Key Steps in Simplifying Expressions:

• Apply the distributive property where necessary.
• Combine like terms by adding or subtracting them.
• Rewrite the simplified expression with the fewest terms possible.

Distributive Property

The distributive property is a cornerstone of algebra that allows us to multiply a single term by each term within a parenthesis. It's often written as \(a(b + c) = ab + ac\), stating that you distribute the multiplication of \(a\) to both \(b\) and \(c\).

When we see expressions like \(5 \times (x - 1)\), we apply the distributive property by multiplying 5 with \(x\) to get \(5x\), and then multiplying 5 with \(–1\) to get \(–5\). The distributive property helps make expressions simpler to work with and is particularly useful in eliminating parentheses and combining like terms, a critical step in the simplification process.

Importance of the Distributive Property:

• It allows simplification of expressions that contain parentheses.
• It helps to identify and combine like terms more readily.
• It is essential in algebraic manipulation and solving equations.