Alright, let's solve the equation step-by-step:
[tex]\[ 0.6 + 2.5x - \frac{1}{6} = -\frac{1}{2} \][/tex]
### Step 1: Simplify the constants on the left-hand side
First, let's combine the constant terms [tex]\(0.6\)[/tex] and [tex]\(-\frac{1}{6}\)[/tex].
Convert [tex]\(0.6\)[/tex] to a fraction for easier calculations:
[tex]\[ 0.6 = \frac{6}{10} = \frac{3}{5} \][/tex]
Now, convert [tex]\(\frac{3}{5}\)[/tex] to have a common denominator with [tex]\(\frac{1}{6}\)[/tex]:
[tex]\[
\frac{3}{5} \text{ and } \frac{1}{6}
\][/tex]
The least common multiple of 5 and 6 is 30. Let's convert the fractions [tex]\(\frac{3}{5}\)[/tex] and [tex]\(\frac{1}{6}\)[/tex] to have the denominator of 30:
[tex]\[
\frac{3}{5} = \frac{3 \cdot 6}{5 \cdot 6} = \frac{18}{30}
\][/tex]
[tex]\[
\frac{1}{6} = \frac{1 \cdot 5}{6 \cdot 5} = \frac{5}{30}
\][/tex]
Now, subtract these fractions:
[tex]\[
\frac{18}{30} - \frac{5}{30} = \frac{18 - 5}{30} = \frac{13}{30}
\][/tex]
So the equation now looks like:
[tex]\[
\frac{13}{30} + 2.5x = -\frac{1}{2}
\][/tex]
### Step 2: Eliminate the fraction on the right-hand side
Convert [tex]\(-\frac{1}{2}\)[/tex] to have the same denominator as [tex]\(\frac{13}{30}\)[/tex]:
[tex]\[
-\frac{1}{2} = -\frac{1 \cdot 15}{2 \cdot 15} = -\frac{15}{30}
\][/tex]
### Step 3: Isolate the term with [tex]\( x \)[/tex]
The equation is now:
[tex]\[
\frac{13}{30} + 2.5x = -\frac{15}{30}
\][/tex]
Subtract [tex]\(\frac{13}{30}\)[/tex] from both sides:
[tex]\[
2.5x = -\frac{15}{30} - \frac{13}{30}
\][/tex]
Combine the fractions on the right side:
[tex]\[
2.5x = -\frac{15 + 13}{30} = -\frac{28}{30}
\][/tex]
Simplify [tex]\(-\frac{28}{30}\)[/tex]:
[tex]\[
2.5x = -\frac{14}{15}
\][/tex]
### Step 4: Solve for [tex]\( x \)[/tex]
Divide both sides by 2.5 to isolate [tex]\( x \)[/tex]:
[tex]\[
x = -\frac{14}{15} \div 2.5
\][/tex]
Convert 2.5 to a fraction:
[tex]\[
2.5 = \frac{5}{2}
\][/tex]
So the equation becomes:
[tex]\[
x = -\frac{14}{15} \div \frac{5}{2} = -\frac{14}{15} \cdot \frac{2}{5} = -\frac{14 \cdot 2}{15 \cdot 5} = -\frac{28}{75}
\][/tex]
Converting [tex]\(-\frac{28}{75}\)[/tex] to a decimal gives approximately [tex]\(-0.373333333333333\)[/tex].
Thus, the solution to the equation is:
[tex]\[
x \approx -0.373333333333333
\][/tex]
