Problem 30 - Precalculus Student Solutions Manual 5th

Question

Determine whether each equation is an identity, a conditional equation, or \(a\) contradiction. Give the solution set. See Example 3 . $$\frac{1}{2}(6 x+20)=x+4+2(x+3)$$

Step-by-Step Solution

Answer

The equation is an identity with the solution set \(\text{R}\).

Step 1 - Distribute on the left-hand side

Distribute \(\frac{1}{2}\) to both terms inside the parentheses: \(\frac{1}{2}(6x + 20)\). This simplifies to \(3x + 10\). So, the equation becomes: \(3x + 10 = x + 4 + 2(x + 3)\).

Step 2 - Simplify the right-hand side

Simplify the terms on the right side by first distributing 2 to both terms inside the parentheses: \(2(x + 3)\) becomes \(2x + 6\). Thus, the equation becomes: \(3x + 10 = x + 4 + 2x + 6\).

Step 3 - Combine like terms on the right-hand side

Combine like terms on the right-hand side: \(x + 4 + 2x + 6\) simplifies to \(3x + 10\). So the equation is now \(3x + 10 = 3x + 10\).

Step 4 - Determine the type of equation

Since both sides of the equation are equal \(3x + 10 = 3x + 10\), this indicates that the equation is an identity. An identity is true for all values of \(x\).

Step 5 - State the solution set

Since the equation is an identity, the solution set is all real numbers, denoted as \(\text{R}\).

Key Concepts

identity equation

An identity equation is a type of equation that holds true for all possible values of the variable. In other words, no matter what value you substitute for the variable, the equation will always be correct. For example, the equation \(3x + 10 = 3x + 10\) is an identity. You can replace \(x\) with any number, and the equation will still be valid. Identity equations indicate that the expressions on both sides are essentially equivalent to each other. As a result, the solution set for an identity equation is all real numbers, denoted as \(\text{R}\). This means that there are no restrictions on the variable for the equation to hold true.

conditional equation

A conditional equation, in contrast to an identity equation, is an equation that is only true for specific values of the variable. This means there are one or more particular numbers that satisfy the equation, but not every number will work. For instance, if we have the equation \(2x + 3 = 7\), you can solve for \(x\) and find that \(x = 2\). However, no other number for \(x\) will make this equation true. Conditional equations are essential in problem-solving because they help us find precise values where certain conditions are met. To find the solution set of a conditional equation, you generally need to isolate the variable on one side of the equation.

contradiction

A contradiction is an equation that has no possible solutions. This means that no matter what you substitute for the variable, the equation will never be true. For example, consider the equation \(x + 2 = x - 5\). If you attempt to isolate \(x\), you'll end up with an impossible statement like \(2 = -5\), which is clearly false. Contradictions occur when the expressions on both sides of the equation are fundamentally incompatible. In the context of solving equations, arriving at a contradiction means that there is no value of the variable that will satisfy the equation, and therefore the solution set is empty.