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Chapter 4: Problem 47

Simplify the expression. (Skills Review Handbook) \(-(-\mathrm{x})\)

Short Answer

Expert verified
The simplified form of the expression \(-(-x)\) is \(x\).

Step by step solution

01

Identify the double negative

In the expression \(-(-x)\), there is a double negative, which occurs when a negative number is subtracted. It is important to understand that two negatives make a positive. So, when we have \(-(-x)\), it can be interpreted as 'the negative of negative x'.
02

Simplify the expression

Applying the principle that a double negative makes a positive, \(-(-x)\) simplifies to \(x\).

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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 a fundamental aspect of algebra that helps us make complex mathematical statements more manageable. When we simplify an expression, we aim to reduce it to its most basic form without changing its original value. This involves reducing terms and performing operations that lessen the complexity of the expression.
For example, in the phrase \(-(-x)\), simplification helps us interpret and rewrite it in its simplest form, which is \(x\).
Common techniques used in simplification include:
  • Combining like terms to reduce clutter.
  • Applying arithmetic rules such as addition, subtraction, multiplication, and division.
  • Employing properties of operations such as distributive or associative laws.
By mastering these techniques, students can tackle more sophisticated problems with confidence and ease.
Double Negative Property
The double negative property is a specific rule in algebra stating that applying two negatives in an expression results in a positive. This can seem counterintuitive at first, but it stems from basic arithmetic principles.
In algebra, when you encounter a double negative such as \(-(-x)\), you should understand it as 'taking the opposite of the opposite鈥�.
Here's a mental shortcut:
  • If a negative sign is applied twice, they neutralize, resulting in a positive. So \(-(-x)\) simplifies to \(x\).
This property is crucial for working with expressions efficiently and allows you to see patterns and relationships more clearly. Recognizing and applying the double negative property ensures accuracy and proficiency in solving equations that involve multiple operations.
Basic Algebra Concepts
Understanding basic algebra concepts lays the foundation for higher-level mathematics. It involves grasping fundamental operations and principles that govern the manipulation of mathematical symbols.
These concepts include:
  • Variables: Symbols that represent numbers or values in equations.
  • Operations: Basic mathematical procedures such as addition, subtraction, multiplication, and division.
  • Properties of operations: Rules like the commutative, associative, and distributive properties that dictate how operations are performed.
For instance, when simplifying expressions or dealing with double negatives, these core ideas empower students to make sense of and solve problems.
By continuously practicing these skills, students can become more competent in handling various algebraic tasks and prepare themselves for more complex mathematical challenges.

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Most popular questions from this chapter

In Exercises \(17-20,\) graph \(\Delta\) RST with vertices \(R(4,1)\) \(S(7,3),\) and \(T(6,4)\) and its image after the glide relection. $$ \begin{array}{l}{\text { Translation: }(\mathrm{x}, \mathrm{y}) \rightarrow(\mathrm{x}, \mathrm{y} | \text { 4) }} \\ {\text { Re ection: in the line } \mathrm{x}=3}\end{array} $$

CONSTRUCTION Follow these steps to construct a rotation of \(\Delta\) ABC by angle \(D\) around a point \(O .\) Use a compass and straightedge. Step 1 Draw \(\Delta \mathrm{ABC}, \angle \mathrm{D}\) , and \(\mathrm{O},\) the center of rotation. Step 2 \(\mathrm{Draw} \overline{\mathrm{OA}}\) Use the construction for copying an angle to copy \(\angle \mathrm{D}\) at \(\mathrm{O},\) as shown. Then use distance OA and center \(\mathrm{O}\) to \(\mathrm{} \mathrm{A}^{\prime}\) Step 3 Repeat Step 2 to Ind points \(B^{\prime}\) and \(C^{\prime} .\) Draw \(\Delta A^{\prime} B^{\prime} C^{\prime}\)

Solve the equation. $$-2(8-y)=-6 y$$

In Exercises \(13-16,\) use the translation. $$ (\mathrm{x}, \mathrm{y}) \rightarrow(\mathrm{x} \mp 8, \mathrm{y}+4) $$ What is the image of \(\mathrm{A}(2,6) ?\)

Determine whether the polygons with the given vertices are similar. Use transformations to explain your reasoning. \(\mathrm{D}(-4,3), \mathrm{E}(-2,3), \mathrm{F}(-1,1), \mathrm{G}(-4,1)\) and \(\mathrm{L}(1,-1), \mathrm{M}(3,-1), \mathrm{N}(6,-3), \mathrm{P}(1,-3)\)
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