Problem 43 - Algebra for College Students

Question

Solve each equation, and check the solution. If applicable, tell whether the equation is an identity or a contradiction. \(2[x-(2 x+4)+3]=2(x+1)\)

Step-by-Step Solution

Answer

The solution is \(x = -1\), and the equation is neither an identity nor a contradiction.

Step 1: Simplify Inside the Brackets

Start by simplifying the expression inside the brackets.Given equation: \[2[x-(2x+4)+3]=2(x+1)\]Simplify inside the brackets: \[x-(2x+4)+3 = x-2x-4+3 = -x-1\]The equation now looks like: \[2[-x-1]=2(x+1)\]

Step 2: Distribute the 2

Distribute the 2 on both sides of the equation.Left side: \[2(-x-1) = -2x-2\]Right side remains: \[2(x+1) = 2x+2\]The equation now looks like: \[-2x-2=2x+2\]

Step 3: Combine Like Terms

Get all the x terms on one side and constants on the other.Add \(2x\) to both sides:\[-2x+2x-2=2x+2x+2\]Simplify: \[-2=4x+2\]Subtract 2 from both sides: \[-2 - 2 = 4x + 2 - 2\]Simplify: \[-4 = 4x\]

Step 4: Solve for x

Divide by 4 on both sides to solve for x:\[\frac{-4}{4} = \frac{4x}{4}\]Simplify to find:\[x = -1\]

Step 5: Check the Solution

Substitute \(x = -1\) back into the original equation to verify the solution.Original equation: \[2[x-(2x+4)+3]=2(x+1)\]Substitute \(x = -1\): \[2[(-1)-(2(-1)+4)+3]=2(-1+1)\]Simplify inside the brackets: \[(-1)-(2(-1)+4)+3 = -1+2-4+3 = 0\]Left side becomes: \[2[0]=0\]Right side: \[2(0)=0\]Both sides are equal, confirming that \(x = -1\) is correct.

Step 6: Determine Identity or Contradiction

This equation has a single solution, which is \(x = -1\), so it is neither an identity nor a contradiction.

Key Concepts

Distributive Property

The distributive property helps us to simplify algebraic expressions. It's a way to remove parentheses by distributing a multiplied value across the terms inside the parentheses. Here’s an example from our equation:
Given: \ \[2[x-(2x+4)+3]=2(x+1)\]
First, we simplify inside the brackets: \[x-(2x+4)+3 = x-2x-4+3 = -x-1\]
Now, we apply the distributive property to the left side: \[2(-x-1) = -2x-2\]
This means multiplying 2 by each term within the parentheses, resulting in \[2 \times -x + 2 \times -1 = -2x-2\]
Remember, the key is distributing the number (2 in this case) to both terms inside the brackets.

Combining Like Terms

Combining like terms means to simplify an expression by merging terms that have the same variable raised to the same power. Let's see this in our example: \[ -2x-2=2x+2\]
To isolate the variable, we need all x terms on one side. Add 2x to both sides: \[ -2x + 2x - 2 = 2x + 2x + 2\]
This combines the x terms: \[ -2 = 4x + 2\]
Next, we combine constants by subtracting 2 from both sides: \[ -4 = 4x\]
We now have a simplified linear equation where like terms are combined on either side.

Checking Solutions

It's crucial to verify that your solution is accurate by substituting it back into the original equation. Here's how:
Original equation: \[2[x-(2x+4)+3]=2(x+1)\]
Our solution for x is -1. Substitute x = -1: \[2[(-1)-(2(-1)+4)+3]=2(-1+1)\]
Simplify inside the brackets: \[ -1 - ( -2 + 4 ) + 3 = -1 + 2 - 4 + 3 = 0\]
So, the left side of the equation is: \[2 \times 0 = 0\]
The right side remains 0 as well: \[2 \times 0 = 0\]
Both sides are equal, which confirms that x = -1 is the correct solution. Always perform this step to ensure accuracy.

Simplifying Expressions

Simplifying expressions involves making them as streamlined as possible by using arithmetic and algebraic properties. Let's walk through the steps using our example: \[ 2[x - (2x + 4) + 3] = 2(x + 1)\]
Start by simplifying inside the brackets: \[ x - 2x - 4 + 3 = -x - 1\]
The equation now is: \[ 2(-x - 1) = 2(x + 1)\]
Distribute the 2: \[ -2x - 2 = 2x + 2\]
To simplify further, get all x terms on one side and constants on the other. Add 2x to both sides: \[ -2 = 4x + 2\]
Subtract 2 from both sides: \[ -4 = 4x\]
Finally, solve for x: \[ x = -1\]
Simplifying expressions makes equations easier to handle and solve. Always work step-by-step to avoid mistakes.