Question
VARIABLE EQUATIONS & INEQUALITIES UNIT TEST:
JUSTIFY YOUR STEPS USING THE PROPERTIES OF EQUALITY AND DETERMINE THE NUMBER OF SOLUTIONS TO THE EQUATION.
\frac{3}{2}\left(4-8x\right)+5x=1-7x
JUSTIFY YOUR STEPS USING THE PROPERTIES OF EQUALITY AND DETERMINE THE NUMBER OF SOLUTIONS TO THE EQUATION.
\frac{3}{2}\left(4-8x\right)+5x=1-7x
Answers
GPT-4o mini
To solve the equation
\[
\frac{3}{2}(4 - 8x) + 5x = 1 - 7x,
\]
let's follow the steps and justify them using properties of equality.
### Step 1: Distribute
Distribute \(\frac{3}{2}\) across \(4 - 8x\):
\[
\frac{3}{2}(4) - \frac{3}{2}(8x) + 5x = 1 - 7x.
\]
Calculating the distribution:
\[
6 - 12x + 5x = 1 - 7x.
\]
### Step 2: Combine Like Terms
Combine like terms on the left side:
\[
6 - 7x = 1 - 7x.
\]
### Step 3: Add \(7x\) to Both Sides
To isolate terms, add \(7x\) to both sides:
\[
6 - 7x + 7x = 1 - 7x + 7x,
\]
which simplifies to
\[
6 = 1.
\]
### Step 4: Analyze the Result
The equation \(6 = 1\) is a contradiction and is never true.
### Conclusion
Since we reached a contradiction, there are **no solutions** to the equation. In summary, the result implies that there are no values of \(x\) that satisfy the given equation.
\[
\frac{3}{2}(4 - 8x) + 5x = 1 - 7x,
\]
let's follow the steps and justify them using properties of equality.
### Step 1: Distribute
Distribute \(\frac{3}{2}\) across \(4 - 8x\):
\[
\frac{3}{2}(4) - \frac{3}{2}(8x) + 5x = 1 - 7x.
\]
Calculating the distribution:
\[
6 - 12x + 5x = 1 - 7x.
\]
### Step 2: Combine Like Terms
Combine like terms on the left side:
\[
6 - 7x = 1 - 7x.
\]
### Step 3: Add \(7x\) to Both Sides
To isolate terms, add \(7x\) to both sides:
\[
6 - 7x + 7x = 1 - 7x + 7x,
\]
which simplifies to
\[
6 = 1.
\]
### Step 4: Analyze the Result
The equation \(6 = 1\) is a contradiction and is never true.
### Conclusion
Since we reached a contradiction, there are **no solutions** to the equation. In summary, the result implies that there are no values of \(x\) that satisfy the given equation.