Let's solve the problem step-by-step:
1. Understand the problem:
We need to find a number such that when we multiply it by 6 and then add 4, the result is equal to taking twice the number subtracted from 52.
2. Define the variable:
Let [tex]\( x \)[/tex] be the number we're looking for.
3. Set up the equation:
According to the problem, the product of 6 and [tex]\( x \)[/tex], increased by 4, is equal to 52 minus twice the number.
This translates to the equation:
[tex]\[
6x + 4 = 52 - 2x
\][/tex]
4. Solve the equation for [tex]\( x \)[/tex]:
First, let's move all the terms involving [tex]\( x \)[/tex] to one side of the equation, and the constant terms to the other side.
Add [tex]\( 2x \)[/tex] to both sides:
[tex]\[
6x + 2x + 4 = 52
\][/tex]
Which simplifies to:
[tex]\[
8x + 4 = 52
\][/tex]
Next, subtract 4 from both sides:
[tex]\[
8x = 48
\][/tex]
Finally, divide both sides by 8:
[tex]\[
x = 6
\][/tex]
5. Check the solution:
Substitute [tex]\( x = 6 \)[/tex] back into the original problem to ensure it satisfies the condition.
- The product of 6 and the number [tex]\( x \)[/tex] is [tex]\( 6 \times 6 = 36 \)[/tex].
- Increased by 4: [tex]\( 36 + 4 = 40 \)[/tex].
On the other side:
- Twice the number is [tex]\( 2 \times 6 = 12 \)[/tex].
- Subtracted from 52: [tex]\( 52 - 12 = 40 \)[/tex].
Both sides of the equation match, confirming that our solution is correct.
Therefore, the number is [tex]\( \boxed{6} \)[/tex].
