We start with the equation:
\[ \frac{2}{3}x - 5 = 10x + 7 \]
To isolate \( x \), we first add 5 to both sides:
\[ \frac{2}{3}x = 10x + 7 + 5 \]
This simplifies to:
\[ \frac{2}{3}x = 10x + 12 \]
Next, we can subtract \( 10x \) from both sides:
\[ \frac{2}{3}x - 10x = 12 \]
To combine the terms, we need a common denominator. The common denominator between \( \frac{2}{3} \) and the whole number \( 10 \) (which can be expressed as \( \frac{30}{3} \)) is 3. Therefore, we rewrite \( 10x \):
\[ \frac{2}{3}x - \frac{30}{3}x = 12 \]
This simplifies to:
\[ \frac{2 - 30}{3}x = 12 \]
Which simplifies further to:
\[ \frac{-28}{3}x = 12 \]
To solve for \( x \), multiply both sides by the reciprocal of \( \frac{-28}{3} \):
\[ x = 12 \times \frac{-3}{28} \]
Calculating the right side:
\[ x = \frac{-36}{28} \]
This can be simplified:
\[ x = \frac{-9}{7} \]
Now we convert \( \frac{-9}{7} \) to decimal form:
\[ x \approx -1.2857142857142857 \]
Rounding to the nearest hundredth:
\[ x \approx -1.29 \]
Thus, the solution is:
\[ \boxed{-1.29} \]
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