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Chapter 4: Problem 49

Simplify the expression. (Skills Review Handbook) \(x-(12-5 x)\)

Short Answer

Expert verified
The simplified expression is \(6x - 12\).

Step by step solution

01

Distribute the negative

The first parenthesis is multiplied by negative 1, so the expression becomes \(x-12+5x\),
02

Combine like terms

Then we combine the similar terms. The 'x' and the '5x' are like terms, so when you add them together, the expression becomes \(6x - 12\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Expression Simplification
Expression simplification is like cleaning up a room of mathematical terms. You want to make the expression as compact and neat as possible. This process helps by reducing complex problems into easier forms, making them more manageable to solve.
When simplifying expressions:
  • Look for opportunities to combine terms.
  • Identify terms that can be factored out or canceled.
  • Consider any rules like the Distributive Property or the Associative Property.
A simplified expression is reduced to its most basic components while still retaining its original value. This is crucial for solving algebra problems because it allows you to quickly grasp the size and shape of what you're working with.
Distributive Property
The Distributive Property is a handy tool in algebra that allows you to multiply a number by a group of numbers added together, by distributing the multiplication to each term inside the parentheses. It's often written as:
\( a(b + c) = ab + ac \)
Imagine you want to simply distribute gifts to friends; you give each their own portion rather than handing everything to one person. The same stands for numbers.
  • This property aids in removing parentheses in expressions, which is a big step towards simplification.
  • It shows that multiplication can "spread out" over addition or subtraction.
In the context of our exercise, we used the Distributive Property to distribute a negative sign across the terms inside the parenthesis. As a result, the expression turned from \(x - (12 - 5x)\) to \(x - 12 + 5x\).
Combining Like Terms
Combining like terms is a fundamental part of simplifying expressions. It involves merging terms with the same variable raised to the same power. This is akin to consolidating similar items in a shopping cart; you group similar products together to see your total more clearly.
  • The terms must have identical variables and exponents. This means that \(3x\) and \(2x\) can be combined, but \(3x\) and \(3x^2\) cannot.
  • It reduces the expression to the smallest number of terms while preserving its overall value.
In our problem, combining \(x\) and \(5x\) (like terms) resulted in \(6x\), giving us the expression \(6x - 12\). By combining like terms, algebra becomes less cluttered and equations become easier to solve.

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