Problem 2 - Prealgebra

Question

Translate each sentence into an equation. Then find each number. Four less than twice a number is \(-2\)

Step-by-Step Solution

Answer

The number is 1.

Step 1: Understand the Sentence

The sentence 'Four less than twice a number is -2' needs to be broken down into parts. We need to identify the mathematical expressions that represent 'four less than twice a number' and 'is -2.'

Step 2: Translate the Words into Expressions

Let the unknown number be represented by the variable \( x \). The phrase 'twice a number' becomes \( 2x \). 'Four less than twice a number' becomes \( 2x - 4 \).

Step 3: Formulate the Equation

Based on our translation, the sentence 'Four less than twice a number is -2' converts to the equation: \[ 2x - 4 = -2 \]

Step 4: Solve the Equation

To find the value of \( x \), solve the equation. Start by adding 4 to both sides:\[ 2x - 4 + 4 = -2 + 4 \]which simplifies to \[ 2x = 2 \].Next, divide both sides by 2:\[ x = 1 \].

Step 5: Verify the Solution

Substitute \( x = 1 \) back into the original expression \( 2x - 4 \) to check:\[ 2(1) - 4 = 2 - 4 = -2 \].The equation is satisfied, confirming our solution is correct.

Key Concepts

Equations

Equations are fundamental to algebra and mathematics in general. They are statements that express the equality of two mathematical expressions. In our exercise, we use an equation to express a relationship described in words.

• In equations, we usually have variables—symbols that stand in for unknown numbers.
• An equation consists of two sides separated by an equal sign (=), indicating that both sides are balanced or equal to each other.
• Understanding how to translate words into equations is a critical skill in algebra. This allows us to represent and solve real-world problems mathematically.

Consider the phrase: 'Four less than twice a number is -2'. Here, 'twice a number' means we double the number, represented by \(2x\). 'Four less than' means we then subtract 4 from \(2x\), which captures the entire phrase as \(2x - 4\). Thus, we get the equation \(2x - 4 = -2\).

Solving Equations

Solving equations involves finding the value of the variable that balances the equation and makes it true.
To solve equations like the one we have \(2x - 4 = -2\):

• Perform operations to isolate the variable on one side of the equation.
• Regularly use inverse operations to simplify equations, such as adding or subtracting the same number on both sides.
• For this equation, add 4 to both sides to eliminate the \(-4\): \(2x - 4 + 4 = -2 + 4\). This simplifies to \(2x = 2\).
• Finally, divide both sides by 2 to solve for \(x\): \(x = 1\).
• Always verify your solution by substituting it back into the original equation to ensure both sides are equal.

These steps demonstrate how equations can be solved systematically to find unknown values.

Prealgebra

Prealgebra is a vital stage in mathematics that prepares students for algebra. It focuses on basic arithmetic and introduces fundamental concepts like variables, expressions, and equations.

• In prealgebra, students learn to work with numbers in new ways, including how to apply operations to variables like \(x\).
• Problems typically involve translating everyday phrases into mathematical equations, as demonstrated in the exercise 'Four less than twice a number is -2'.
• This prealgebra problem introduces the concept of equations and demonstrates balancing equations through logical thinking and operations.
• By understanding how to develop and solve such equations, students build a strong foundation for more advanced algebraic concepts.

Mastering prealgebra equips students with skills and confidence, paving the way for success in more complex areas of math.