Problem 8 - Algebra 1

Question

Match the verbal phrase with its corresponding algebraic expression. A. \(4 x-11\) B. \(4(x-11)\) c. \(11-4 x\) D. \(11 x+4\) Four times a number \(x\) decreased by eleven

Step-by-Step Solution

Answer

The correct algebraic expression for the phrase 'Four times a number \(x\) decreased by eleven' is \(4x-11\), which is answer A.

Step 1: Interpret the phrase

The first step is to break down the phrase 'Four times a number \(x\) decreased by eleven' into math operations. The word 'times' signifies multiplication and 'decreased by' signifies subtraction. Thus, the phrase shows multiplication first, followed by subtraction.

Step 2: Match the phrase with the expressions

After interpreting the phrase, the resulting algebraic expression should be 'four times a number \(x\)' which is \(4x\), 'decreased by eleven' will be \(-11\). Now look for the algebraic expression equivalent, which is \(4x-11\).

Step 3: Find the correct match

Look for the algebraic expression \(4x-11\) in the given answers. The correct match is answer A, \(4x-11\).

Key Concepts

Verbal Phrases

In algebra, converting verbal phrases into algebraic expressions is a fundamental skill. Verbal phrases are sentences that describe mathematical operations using words. For example, a phrase like "Four times a number \(x\) decreased by eleven" uses everyday language to express a mathematical idea. Understanding this language helps us translate words into symbols, allowing us to solve problems more effectively.

Let's break down the given example:

• "Four times" indicates multiplication. So, if \(x\) is the unknown number, "four times \(x\)" becomes \(4x\).
• "Decreased by eleven" tells us to subtract eleven. Hence, from \(4x\), we subtract \(11\), giving us \(4x - 11\).

When interpreting verbal phrases, look for keywords: 'times' usually means to multiply, while 'decreased by' means to subtract. Identifying these keywords is crucial for making accurate translations to algebraic form.

This skill not only aids in algebra problems but also in real-life scenarios where calculations are described verbally.

Multiplication and Subtraction

Once a verbal phrase has been understood, the next step is implementing the operations—like multiplication and subtraction—in algebraic expressions. These operations are the backbone of many mathematical problems and equations.

• **Multiplication**: The phrase "Four times a number \(x\)" becomes \(4x\). Here, \(4\) is the coefficient multiplying the variable \(x\).
• **Subtraction**: We express "decreased by eleven" as subtracting the number \(11\). Hence, \(4x\) turning into \(4x - 11\).

The order of operations is important. Multiplication is performed before subtraction because the phrase indicates that the product of four and \(x\) is calculated first, followed by reducing the result by eleven.

Misunderstanding these operations can lead to incorrect results. By practicing, you will grow comfortable interpreting and implementing them, which is essential for solving problem sets effectively.

Matching Expressions

After understanding and implementing operations from a verbal phrase, the next task is to match the result with the correct algebraic expression. This involves comparing several expressions to find the one that represents the intended calculations correctly.

Let's consider our example: "Four times a number \(x\) decreased by eleven". We derived the expression \(4x - 11\). When given a list of potential matches:

• Look at each expression one by one.
• Identify the expression that shows the same operations, in the same order, as your derived expression.

Here, after breaking down and implementing the phrase, the correct match amongst the options \(4x-11\), \(4(x-11)\), \(11-4x\), and \(11x+4\) is option A, which is \(4x - 11\).

This process of comparison and matching is a vital skill in algebra, bolstering one’s ability to ensure accuracy in assignments and tests. It also sharpens your critical thinking, helping you check and cross-reference your work effectively.