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Chapter 2: Problem 50

Solve each equation, and check the solution. \(4(x+3)=2(2 x+8)-4\)

Short Answer

Expert verified
The equation is an identity, and any value of \(x\) is a solution.

Step by step solution

01

- Simplify Both Sides

Start by distributing the numbers outside the parentheses on both sides of the equation. For the left side, distribute 4 to both terms inside the parentheses, and for the right side, distribute 2 to both terms inside the parentheses.Left side: \[4(x+3) = 4x + 12 \]Right side: \[2(2x + 8) = 4x + 16\]
02

- Combine Like Terms

After distributing, our equation looks like this: \[4x + 12 = 4x + 16 - 4\]Combine like terms on the right side:\[4x + 12 = 4x + 12\]
03

- Isolate the Variable

Subtract \(4x\) from both sides to isolate the variable:\[4x + 12 - 4x = 4x + 12 - 4x\]This simplifies to:\[12 = 12\]
04

- Check the Solution

Since both sides of the equation are equal, any value of \(x\) will satisfy the equation. The equation is an identity and has infinitely many solutions.

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Key Concepts

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

Distributive Property
The distributive property allows you to multiply a term outside the parentheses by each term inside the parentheses. This is crucial for simplifying equations. In our exercise, we apply the distributive property as follows:

  • For the left side: Distribute 4 to both inside terms: \(4(x+3) = 4x + 12\)

  • For the right side: Distribute 2 to both inside terms: \(2(2x + 8) = 4x + 16\)

Applying the distributive property helps us expand and simplify each side of the equation, making it easier to solve in subsequent steps.
Combining Like Terms
Once we've used the distributive property, we often need to combine like terms. Like terms are terms that contain the same variable raised to the same power. In our problem, this step occurs as follows:

  • After distributing, we form: \(4x + 12 = 4x + 16 - 4\)

Combining the constant terms on the right side, we get:
  • \(4x + 12 = 4x + 12\)
This simplification makes it clearer to see if the equation has a solution or if further steps are necessary.
Isolating the Variable
Isolating the variable means manipulating the equation so that the variable is alone on one side of the equation, which is necessary for solving for the variable. Here鈥檚 how we isolate the variable in our example:

  • After combining like terms, we have \(4x + 12 = 4x + 12\)
  • Subtract 4x from both sides: \(4x + 12 - 4x = 4x + 12 - 4x\)

This simplifies to:

  • \(12 = 12\)
Since isolating the variable led to a true statement, it tells us something important about the type of equation we are dealing with.
Identity Equation
An identity equation is a type of equation that is true for all values of the variable. After simplifying and isolating the variable, we are left with:

  • \(12 = 12\)

This indicates that the original equation is an identity. This means any value of \(x\) will satisfy the equation. Understanding identity equations helps us recognize when an equation has infinite solutions, making it a powerful concept in algebra.

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