Sure, let's verify the given equation step by step.
### Step 1: Simplify Inside the Brackets
First, we need to simplify the expression inside the brackets on the left-hand side:
[tex]\[
\frac{3}{7} + \frac{-12}{5}
\][/tex]
To add these two fractions, we need a common denominator. The least common multiple of 7 and 5 is 35.
So, we convert each fraction:
[tex]\[
\frac{3}{7} = \frac{3 \times 5}{7 \times 5} = \frac{15}{35}
\][/tex]
[tex]\[
\frac{-12}{5} = \frac{-12 \times 7}{5 \times 7} = \frac{-84}{35}
\][/tex]
Now, we add the two fractions:
[tex]\[
\frac{15}{35} + \frac{-84}{35} = \frac{15 - 84}{35} = \frac{-69}{35}
\][/tex]
### Step 2: Multiply with the Fraction Outside the Brackets
Now, multiply this result by [tex]\(\frac{-15}{4}\)[/tex]:
[tex]\[
\frac{-15}{4} \times \frac{-69}{35}
\][/tex]
Multiplying these fractions:
[tex]\[
\frac{-15 \times -69}{4 \times 35} = \frac{1035}{140}
\][/tex]
### Step 3: Simplify the Result
Simplify the fraction [tex]\(\frac{1035}{140}\)[/tex]:
Both 1035 and 140 are divisible by 5:
[tex]\[
\frac{1035 \div 5}{140 \div 5} = \frac{207}{28} \approx 7.392857142857142
\][/tex]
### Step 4: Verify the Right-Hand Side
First, perform the multiplications inside the brackets:
First Term:
[tex]\[
\frac{-15}{4} \times \frac{3}{7} = \frac{-15 \times 3}{4 \times 7} = \frac{-45}{28} \approx -1.6071428571428572
\][/tex]
Second Term:
[tex]\[
\frac{-15}{4} \times \frac{-12}{5} = \frac{-15 \times -12}{4 \times 5} = \frac{180}{20} = 9.0
\][/tex]
### Step 5: Add the Two Terms
Add these two results:
[tex]\[
\frac{-45}{28} + 9 = -1.6071428571428572 + 9 = 7.392857142857143
\][/tex]
### Conclusion
We have verified that both sides of the equation are equal:
[tex]\[
\frac{-15}{4} \times\left[\frac{3}{7}+\frac{-12}{5}\right] \approx 7.392857142857142
\][/tex]
and
[tex]\[
\left[\frac{-15}{4} \times \frac{3}{7}\right]+\left[\frac{-15}{4} \times \frac{-12}{5}\right] \approx 7.392857142857143
\][/tex]
Hence, the equation holds true.
