Verify the following:

(i) [tex]\[\frac{-15}{4} \times\left(\frac{3}{7}+\frac{-12}{5}\right)=\left(\frac{-15}{4} \times \frac{-3}{7}\right)+\left(\frac{-15}{4} \times \frac{-12}{5}\right)\][/tex]



Answer :

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.