Problem 74 - College Algebra Essentials

Question

Combine the types of equations we have discussed in this section. Solve each equation. Then state whether the equation is an identity, a conditional equation, or an inconsistent equation. \(2(x+2)+2 x=4(x+1)\)

Step-by-Step Solution

Answer

The given equation is an identity.

Step 1: Simplify the Equation

Start by distributing \(2\) in both $2*(x+2)$ and $4*(x+1)$, which gives the equation as \(2x+4+2x=4x+4\). Now, simplify this to get the equation: \(4x+4=4x+4\).

Step 2: Subtract '4x + 4' from Both Sides

After that, we subtract \(4x+4\) from both sides of the equation, which gives \(0=0\). This equation is always true.

Step 3: Classify the Equation

At last, classify the equation. The equation is true for all real numbers; thus, it is an identity.

Key Concepts

identity equation

An identity equation is a type of algebraic equation that is true for every possible value of its variables. To better understand this, think of identity equations as statements that maintain their truth no matter what number you replace the variable with. For example, if you encounter an equation like \( 5 + x = x + 5 \), it holds true whether \( x = 1 \), \( x = 10 \), or any other number.

In the given problem \( 2(x+2)+2x=4(x+1) \), when simplified, it becomes \( 4x+4=4x+4 \). This is a clear signature of an identity equation. Since both sides of the equation are identical, it suggests that the equation is satisfied for all values of \( x \).

• Identity equations often result in statements like \( 0 = 0 \) after simplification.
• They aren't particularly useful for finding a specific solution since every number is a solution.
• Recognizing identity equations can save time because they don't require further solving.

Understanding identity equations helps in discerning if an equation has universal solutions or specific conditions.

conditional equation

A conditional equation is true for only certain values of the variable. Unlike identity equations, they don't hold for every number you plug into them.

Consider a simple equation like \( 2x + 3 = 7 \). Here, we can't say it's true for every possible \( x \), because simplifying it leads to a specific solution: \( x = 2 \). This represents a 'condition' that makes the equation true.

• Conditional equations often have one or more specific solutions that satisfy them.
• They require solving techniques like isolating the variable to find the true values.
• Being aware of conditions can help identify specific constraints during problem-solving.

In comparing to the original problem where we find an equation that is identity, the "identity" provides no such specific solutions, indicating it's not merely conditional.

inconsistent equation

Inconsistent equations are the opposite of identity equations. They have no solution because no value of the variable will satisfy the equation. Think of them as conflicting statements where, no matter what number you substitute for the variable, the equation remains false.

For instance, if you come across an equation like \( x + 2 = x - 5 \), you'll find that no number can satisfy this condition. Trying to solve results in a contradiction, for instance, simplifying this equation would lead to an impossible equality like \( 2 = -5 \).

• Inconsistent equations push equation sides into contradictory statements.
• They can result in equations collapsing into statements such as \( 0 = 1 \).
• Recognizing them quickly is useful, allowing you to conclude there are no solutions without further attempts.

While solving equations, it's crucial to identify if you end up with inconsistent scenarios. This not only saves time but also gives insight into the underlying nature of the problem being posed. In our original exercise, however, the result was consistent, making it clear it was not an inconsistent scenario.