Question

1. $|\frac{3x}{4} - 6| = 28$
2. $|\frac{2m + 3}{6}| = 5$
3. $|2y + 3| + 7 = 47$
4. $\frac{0.4}{2w} = \frac{3}{0.75}$
5. $\frac{35}{25} = \frac{15 - 1c}{5}$
6. $\frac{6}{4y + 1} = \frac{9}{9y - 6}$

Explanation:

Let's solve the equation \(\frac{0.4}{2x}=\frac{3}{0.75}\) (from problem 31, assuming the variable is \(x\) instead of \(2x\) maybe a typo, or we proceed as is). Wait, let's check the original: the equation is \(\frac{0.4}{2x}=\frac{3}{0.75}\)? Wait, no, the user's image has \(\frac{0.4}{2x}=\frac{3}{0.75}\)? Wait, let's parse the image: the 31st problem (labeled 31) is \(\frac{0.4}{2x}=\frac{3}{0.75}\)? Wait, no, the numerator on the right is 3? Wait, the image shows \(\frac{0.4}{2x}=\frac{3}{0.75}\)? Wait, maybe it's \(\frac{0.4}{2x}=\frac{3}{0.75}\)? Let's solve it step by step.

Step 1: Cross - multiply

Cross - multiplying the proportion \(\frac{a}{b}=\frac{c}{d}\) gives \(a\times d = b\times c\). For the equation \(\frac{0.4}{2x}=\frac{3}{0.75}\), we have:
\(0.4\times0.75=2x\times3\)

Step 2: Calculate the left - hand side and the right - hand side

First, calculate \(0.4\times0.75\):
\(0.4\times0.75 = 0.3\)
Then, calculate \(2x\times3=6x\)
So our equation becomes \(0.3 = 6x\)

Step 3: Solve for \(x\)

Divide both sides of the equation \(0.3 = 6x\) by 6:
\(x=\frac{0.3}{6}\)
\(x = 0.05\)

Wait, maybe the original equation was \(\frac{0.4}{x}=\frac{3}{0.75}\) (maybe a typo in the denominator as \(x\) instead of \(2x\)). Let's re - solve with denominator \(x\):

Step 1: Cross - multiply

\(0.4\times0.75=x\times3\)

Step 2: Calculate left - hand side

\(0.4\times0.75 = 0.3\)
So \(0.3 = 3x\)

Step 3: Solve for \(x\)

Divide both sides by 3: \(x=\frac{0.3}{3}=0.1\)

Alternatively, if the equation is \(\frac{0.4}{2x}=\frac{3}{0.75}\), the first solution is correct. But let's check the arithmetic again.

If we consider the equation \(\frac{0.4}{2x}=\frac{3}{0.75}\):

Cross - multiply: \(0.4\times0.75=2x\times3\)

\(0.3 = 6x\)

\(x=\frac{0.3}{6}=0.05\)

Another way: Simplify \(\frac{3}{0.75}\) first. \(\frac{3}{0.75}=4\)

So the equation becomes \(\frac{0.4}{2x}=4\)

Multiply both sides by \(2x\): \(0.4 = 4\times2x\)

\(0.4 = 8x\)

\(x=\frac{0.4}{8}=0.05\)

Yes, that's correct.

Answer:

\(x = 0.05\)