Let's go through the steps systematically to solve the inequality given:
Given inequality:
\[ 0.2(x + 20) - 3 > -7 - 6.2x \]
Step 1: Use the distributive property
Distributing \(0.2\) in \(0.2(x + 20)\): \[ 0.2x + 4 - 3 > -7 - 6.2x \]
This simplifies to: \[ 0.2x + 1 > -7 - 6.2x \]
Step 2: Combine like terms
Next, we'll continue the transformation: \[ 0.2x + 1 > -7 - 6.2x \] Now, we want to isolate the variable terms on one side. We'll move \(6.2x\) to the left side.
Step 3: Use the addition property of inequality
Adding \(6.2x\) to both sides: \[ 0.2x + 6.2x + 1 > -7 \] Combining like terms gives: \[ 6.4x + 1 > -7 \]
Step 4: Use the subtraction property of inequality
Now, subtract \(1\) from both sides: \[ 6.4x > -8 \]
Step 5: Use the division property of inequality
Finally, divide both sides by \(6.4\) (note that \(6.4 > 0\), so the direction of the inequality does not change): \[ x > \frac{-8}{6.4} \]
Simplifying \(\frac{-8}{6.4}\): \[ x > -\frac{80}{64} = -\frac{5}{4} = -1.25 \]
Final Answer
Thus, the solution to the inequality is: \[ x > -1.25 \]
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