Problem 72 - Algebra and Trigonometry

Question

Combine the types of equations we have discussed in this section. Solve equation. Then state whether the equation is an identity, a conditional equation, or an inconsistent equation. $$\frac{6}{x+3}+2=\frac{-2 x}{x+3}$$

Step-by-Step Solution

Answer

The solution to the given equation is\(x = -3\), but when substituted back to the original equation, you see that the equation is undefined. Therefore, the given equation is an inconsistent equation because it has no solution.

Step 1: Simplify the Equation

We need to simplify the equation by getting rid of the denominator. To do this, several both sides of the equation by the denominator which is \(x+3\), this gets: \(6 + 2(x+3) = -2x\). The equation simplifies to \(6 + 2x + 6 = -2x\).

Step 2: Rearrange Terms

Rearrange the terms in the equation so that all terms with x are on one side and the constants are on the other side. It becomes \(2x + 2x = -6 -6 \). Simplifying this gives \(4x = -12\).

Step 3: Solve for the Variable x

Divide both sides of the equation by 4 to solve for x. This gives \(x = -12/4\), so \(x = -3\).

Step 4: Identify the type of Equation

Substitute \(x = -3\) back into the original equation. Doing so, we get: \(6/0 + 2 = -2*-3/0\), but any number divided by zero is undefined; which means the original equation is undefined. Therefore, the statement \(x = -3\) does not make the original equation true, thus it's an inconsistent equation because it has no solution.

Key Concepts

Identities and Conditional Equations

When solving rational equations, it's important to understand the difference between identities and conditional equations. An identity is an equation that is true for all values of the variable. For example, the equation \( 3(x+2) = 3x+6 \) simplifies to \( x = x \) which is always true, no matter what value is substituted for \( x \) .

A conditional equation, on the other hand, is only true for certain values of the variable. For instance, the equation \( x + 2 = 5 \) is only true when \( x = 3 \). In the classroom and for homework, students will most often encounter conditional equations. Knowing how to identify the type of equation you are dealing with can greatly help in solving the problem and understanding the possible solutions.

Inconsistent Equations

Some equations, known as inconsistent equations, have no solution. These kinds of equations are false for all values of the variable. The exercise provided shows an inconsistent equation. During the solution process, if substituting the found value for \( x \) leads to a division by zero or another impossibility, it means that the solution is not valid for the original equation. It's essential to check solutions by plugging them back into the original equation to ensure that they do not result in any mathematical inconsistencies, such as dividing by zero. An inconsistent equation might suggest the need for a reassessment of the original problem or context to check for possible errors.

Simplifying Equations

Simplifying equations is a critical step in the process of solving them. It involves reducing the equation to its simplest form, making it easier to identify the solution. To simplify a rational equation, common strategies include clearing fractions, combining like terms, and factoring. In the given exercise, simplifying began with the elimination of the denominator by multiplying both sides by \( x+3 \). Always remember to simplify with caution, as certain actions, like squaring both sides of an equation, can introduce extraneous solutions. Furthermore, simplifying can reveal the nature of the equation, such as indicating an inconsistent equation if you end up with a nonsense statement like \( 0 = 5 \).

Equation Solving Steps

Solving equations involves a number of steps which, if followed carefully, lead to the solution. The steps typically include simplifying the equation, rearranging it to isolate the variable, and performing arithmetic to solve for the variable. Here are the common steps:

• Simplify the equation to remove fractions and combine like terms.
• Rearrange the equation to get all terms containing the variable on one side and constants on the other.
• Perform any necessary operations to solve for the variable. This might be division or factoring, for example.
• Always check your solutions in the original equation to ensure they do not result in undefined expressions or violate any mathematical rules.

Realizing the importance of these steps and applying them meticulously can significantly increase students' success rate in solving equations.