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Chapter 8: Problem 83

Solve. $$ (4 x+15) 1 / 2-2 x=0 $$

Short Answer

Expert verified
The equation has no solution.

Step by step solution

01

Simplify the Expression

Let's simplify the expression by recognizing that the term \((4x + 15) 1/2\) implies multiplication by 1/2. This means we rewrite it as \((4x + 15) / 2\), so the equation becomes:\[\frac{4x + 15}{2} - 2x = 0\]
02

Eliminate the Fraction

To eliminate the fraction, multiply every term in the equation by 2. This gives us:\[4x + 15 - 4x = 0\]
03

Combine Like Terms

Now, notice that \(4x\) and \(-4x\) cancel each other out:\[15 = 0\]Since it simplifies to a statement that is not true, this equation has no solution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Simplifying Expressions
Simplifying expressions is a crucial step when working with algebraic equations. It involves rewriting the expression in a simpler and more understandable form. For instance, in the original problem, the term \((4x + 15) \frac{1}{2}\) might look complex at first.
By recognizing that multiplying by \(\frac{1}{2}\) is equivalent to dividing by 2, we can rewrite the expression as \(\frac{4x + 15}{2}\).
  • This rewritten form makes the expression more manageable.
  • It allows for easier manipulation in subsequent steps, such as eliminating fractions.
Mastering the simplification step can greatly ease the process of solving equations, leading to fewer mistakes and a better understanding of what each term represents.
Solving Equations
Solving equations involves finding the value of the variable that makes the equation true. Once the expression is simplified, the next step is to manipulate the equation to isolate the variable on one side.
In our example, after simplifying, the equation becomes \(\frac{4x + 15}{2} - 2x = 0\). To remove the fraction, multiply every term by 2:
  • This step eliminates the fraction, simplifying the equation further: \(4x + 15 - 4x = 0\).
  • It prepares the equation for combining like terms and solving for the variable.
However, in this particular problem, once we try to solve for \(x\), we notice that the terms \(4x\) and \(-4x\) cancel out, leaving no variable to solve for. This brings us to a unique situation.
No Solution
Sometimes, during the process of solving equations, you might end up with a statement that isn't true. This indicates that the equation has no solution. When solving the example equation, the terms \(4x\) and \(-4x\) cancel each other out, simplifying to \(15 = 0\).
  • This is clearly a contradiction, as 15 does not equal 0.
  • It signifies that no value of \(x\) will satisfy the equation given the conditions laid out in the problem.
Encountering a no-solution scenario can be confusing, but it's a natural part of exploring algebra.
Such equations remind us that sometimes assumptions might be incorrect, or the problem may be set in such a way that no solution exists under the given circumstances.

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