To solve the equation [tex]\(0.25[2.5x + 1.5(x - 4)] = -x\)[/tex], we can follow these steps:
1. Distribute and simplify within the brackets:
[tex]\[
0.25 [2.5x + 1.5(x - 4)]
\][/tex]
First, simplify inside the brackets.
[tex]\[
2.5x + 1.5(x - 4) = 2.5x + 1.5x - 1.5 \cdot 4 = 2.5x + 1.5x - 6
\][/tex]
Combine like terms inside the brackets:
[tex]\[
2.5x + 1.5x - 6 = 4x - 6
\][/tex]
2. Multiply by 0.25:
Now multiply the simplified expression by 0.25:
[tex]\[
0.25 (4x - 6)
\][/tex]
Distribute the 0.25:
[tex]\[
0.25 \cdot 4x - 0.25 \cdot 6 = 1x - 1.5 = x - 1.5
\][/tex]
3. Equate to the right-hand side of the equation:
The equation is now:
[tex]\[
x - 1.5 = -x
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
Move all terms involving [tex]\(x\)[/tex] to one side and constant terms to the other side:
[tex]\[
x + x = 1.5
\][/tex]
Combine like terms:
[tex]\[
2x = 1.5
\][/tex]
Divide both sides by 2 to solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{1.5}{2} = 0.75
\][/tex]
Therefore, the solution to the equation is:
[tex]\[
x = 0.75
\][/tex]
This result shows that [tex]\(x = 0.75\)[/tex] is the value that satisfies the given equation.
