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Chapter 3: Problem 1

Is the equation \(-2(4-x)=2 x-8\) an identity? Explain why or why not.

Short Answer

Expert verified
No, the given equation \(-2(4-x)=2x-8\) is not an identity because the terms order are not the same on both sides.

Step by step solution

01

Simplify the left side of the equation

The left side of the equation is \(-2(4-x)\). By applying the distributive property, the equation simplifies to \(-2*4 - (-2*x)\) which equals \(-8+2x\).
02

Simplify the right side of the equation

The right side of the equation is \(2x - 8\). This equation is already simplified.
03

Compare the simplified left and right sides

After simplification, it can be seen that the left side, \(-8+2x\), matches with the right side, \(2x - 8\). However, because the term of x in the left side and right side is not the same specifically because of terms order, therefore, the equation is not an identity. Terms order is important for the expression to be an identity.

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Key Concepts

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

Understanding the Distributive Property
The distributive property is a fundamental concept in algebra. It's a way of simplifying expressions and making calculations more straightforward. When you see a problem like \(-2(4-x)\), you're dealing with the distributive property. To apply it:
  • Multiply the factor outside the parentheses by each term inside the parentheses.
  • This means \(-2\) is multiplied by both \(4\) and \(-x\).
So for \(-2(4-x)\):- First, multiply \(-2\) by \(4\), which gives \(-8\).- Next, multiply \(-2\) by \(-x\), resulting in \(+2x\).
Combining these, you get \(-8 + 2x\). This simplification step is vital for solving equations, as it helps match expressions on both sides.
The Process of Simplification
Simplifying equations involves reducing them to their simplest form. In our equation, \(-2(4-x)=2x-8\), we start by simplifying the left side. Using the distributive property, we rewrite it as \(-8 + 2x\).
The right side, \(2x - 8\), is already in its simplest form. Simplification is crucial because:
  • It allows easier comparison of both sides of an equation.
  • It helps identify if the equation is an identity or not.
When both sides are simplified, you can directly compare them. In this case, even though the terms \(2x - 8\) and \(-8 + 2x\) involve the same numbers and variables, the differing order indicates that they are not identical due to term arrangement.
Solving the Equation
Equation solving often follows a series of steps to isolate the variable or to understand the relationship between terms. Here, we're checking if \(-2(4-x)=2x-8\) is an identity. After simplifying both sides:
  • You get \(-8 + 2x\) on the left.
  • And \(2x - 8\) on the right.
At first glance, the equations seem similar, containing the same terms. However, the order of terms is crucial in mathematics. For an equation to be an identity:
  • Both sides must be completely equal.
  • All corresponding parts must align identically.
Since \(-8 + 2x\) isn't arranged the same way as \(2x - 8\), this equation isn't an identity. Understanding this distinction helps in determining equations that hold true universally versus those dependent on specific values of variables.

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