Problem 11
Question
Determine whether the equation is an identity, a conditional equation, or a contradiction. $$2(x-1)=2 x-2$$
Step-by-Step Solution
Verified Answer
The equation \(2(x-1)=2x-2\) is an identity because it holds true for all real numbers
1Step 1: Simplify the equation
Rewrite the equation by distributing the multiplication on the left side. This yields \(2x -2 = 2x - 2\).
2Step 2: Compare the two sides
On comparing the two sides, it is observed that they are exactly the same.
3Step 3: Determine the type of equation
Since the equation holds true for all values of \(x\), it is an identity.
Key Concepts
Conditional EquationsContradictions in AlgebraEquation Simplification
Conditional Equations
Understanding conditional equations is crucial for progressing in algebra. A conditional equation is an algebraic equation that is true for certain values of its variables, unlike an identity which is true for all possible values. For example, the equation \(2x + 3 = 11\) is a conditional equation because it is only true when \(x\) equals 4.
To solve a conditional equation, one must perform operations that maintain the equality, such as adding, subtracting, multiplying, or dividing both sides of the equation by the same non-zero number. It is also important to check the solution by substituting it back into the original equation to ensure that it doesn't result in a contradiction such as an impossible statement like \(0=5\).
To solve a conditional equation, one must perform operations that maintain the equality, such as adding, subtracting, multiplying, or dividing both sides of the equation by the same non-zero number. It is also important to check the solution by substituting it back into the original equation to ensure that it doesn't result in a contradiction such as an impossible statement like \(0=5\).
Contradictions in Algebra
Algebraic contradictions are statements that are false for all values of the variable involved. They arise in equations that have no possible solution. An example of a contradiction is the equation \(5x + 2 = 5x - 3\). No matter what value you substitute for \(x\), the equation will never hold true because adding 2 and subtracting 3 will never yield the same result.
Recognizing contradictions is essential when solving algebraic equations. If during the process of solving you arrive at a contradiction, it means that the initial equation has no solution. It's important to understand that equations leading to contradictions are not failures; they simply represent situations that have no viable answer within the set of real numbers.
Recognizing contradictions is essential when solving algebraic equations. If during the process of solving you arrive at a contradiction, it means that the initial equation has no solution. It's important to understand that equations leading to contradictions are not failures; they simply represent situations that have no viable answer within the set of real numbers.
Equation Simplification
Equation simplification is a fundamental skill in algebra that involves manipulating an equation to make it easier to solve. The goal is to express the equation in its simplest form without changing its meaning. This can involve combining like terms, using the distributive property, and canceling out terms.
For instance, the equation from our exercise \(2(x - 1) = 2x - 2\) can be simplified by distributing the multiplication over the parentheses which gives us \(2x - 2 = 2x - 2\). After simplification, it is clear that both sides of the equation mirror each other, indicating that it is an identity since the same quantity is on both sides of the equal sign. Simplifying equations helps identify whether they are conditional, identities, or contradictions.
For instance, the equation from our exercise \(2(x - 1) = 2x - 2\) can be simplified by distributing the multiplication over the parentheses which gives us \(2x - 2 = 2x - 2\). After simplification, it is clear that both sides of the equation mirror each other, indicating that it is an identity since the same quantity is on both sides of the equal sign. Simplifying equations helps identify whether they are conditional, identities, or contradictions.
Other exercises in this chapter
Problem 11
Solve the quadratic equation by factoring. Check your solutions in the original equation. $$x^{2}-10 x+21=0$$
View solution Problem 11
Write the complex number in standard form. $$4+\sqrt{-9}$$
View solution Problem 12
(a) create a scatter plot of the data, (b) draw a line of fit that passes through two of the points, and (c) use the two points to find an equation of the line.
View solution Problem 12
Find all solutions of the equation algebraically. Use a graphing utility to verify the solutions graphically. $$x^{4}-2 x^{3}=16+8 x-4 x^{3}$$
View solution