Solve the Fraction Equation: -1/2(x - 1/4) = 1/2(3 - x)

Q: How can an equation have no solution?

When you solve and the variables completely cancel out, you're left with a statement like $ \frac{1}{8} = \frac{3}{2} $. Since this is never true, no value of x can make the original equation work!

Q: What's the difference between no solution and x = 0?

Very important distinction! When x = 0, that's a real solution you can substitute back. No solution means the equation is impossible - no number will ever work.

Q: Should I keep trying to solve when variables cancel?

No! Once variables cancel completely, stop and look at what's left. If you get something like 2 = 5, that's your answer: no solution exists.

Q: How do I write 'no solution' as an answer?

You can write it as "No solution", "∅" (empty set), or "Inconsistent equation". All mean the same thing - the equation cannot be satisfied.

Linear Equations with No Solution

Solve for X:

$-\frac{1}{2}(x-\frac{1}{4})=\frac{1}{2}(3-x)$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve

00:04 Multiply by 2 to eliminate fractions

00:27 Negative times positive is always negative

00:33 Negative times negative is always positive

00:40 Let's arrange the equation so that one side has only the unknown X

00:49 We got an illogical expression, therefore there is no solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve for X:

$-\frac{1}{2}(x-\frac{1}{4})=\frac{1}{2}(3-x)$

Step-by-step solution

To solve the given equation, follow these steps:

Step 1: Distribute the fractions
For the left side: $-\frac{1}{2}(x-\frac{1}{4}) = -\frac{1}{2}x + \frac{1}{8}$ .
For the right side: $\frac{1}{2}(3-x) = \frac{3}{2} - \frac{1}{2}x$ .
Step 2: Set the distributed expressions equal to each other
Equation: $-\frac{1}{2}x + \frac{1}{8} = \frac{3}{2} - \frac{1}{2}x$ .
Step 3: Simplify and solve for $x$
Notice that the $-\frac{1}{2}x$ terms cancel each other on both sides of the equation:
$\frac{1}{8} = \frac{3}{2}$ .
This clearly is not possible since $\frac{1}{8}$ is not equal to $\frac{3}{2}$ .
Conclusion: Determine if a solution exists
The result indicates a contradiction because both sides of the equation cannot be equal. This implies that no value of $x$ will satisfy the equation.

Therefore, the solution to the problem is that there is no solution.

Final Answer

There is no solution.

Key Points to Remember

Essential concepts to master this topic

Distribution Rule: Apply the distributive property to both sides first
Technique: Cancel like terms: $-\frac{1}{2}x$ cancels from both sides
Check: If constants don't equal, like $\frac{1}{8} ≠ \frac{3}{2}$ , no solution exists ✓

Common Mistakes

Avoid these frequent errors

Assuming all equations have solutions
Don't automatically expect to find an x-value = wrong assumption! When variables cancel and leave unequal constants, this proves no solution exists. Always recognize that $\frac{1}{8} = \frac{3}{2}$ is impossible, meaning no x can satisfy the equation.

Practice Quiz

Test your knowledge with interactive questions

$$ 5x=1 $$

What is the value of x?

FAQ

Everything you need to know about this question

How can an equation have no solution?

When you solve and the variables completely cancel out, you're left with a statement like $\frac{1}{8} = \frac{3}{2}$ . Since this is never true, no value of x can make the original equation work!

What's the difference between no solution and x = 0?

Very important distinction! When x = 0, that's a real solution you can substitute back. No solution means the equation is impossible - no number will ever work.

Should I keep trying to solve when variables cancel?

No! Once variables cancel completely, stop and look at what's left. If you get something like 2 = 5, that's your answer: no solution exists.

How do I write 'no solution' as an answer?

You can write it as "No solution", "∅" (empty set), or "Inconsistent equation". All mean the same thing - the equation cannot be satisfied.

Could I have made an algebra mistake instead?

Always double-check your distribution and arithmetic! But if your work is correct and you get $\frac{1}{8} = \frac{3}{2}$ , then no solution is definitely the right answer.

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