Sure, let's analyze each step carefully to identify any potential errors in the calculations.
### Step-by-Step Solution:
#### Line (1):
Starting with the inequality:
[tex]\[
-2(-5 x + 4) \geq -10
\][/tex]
Let's distribute the [tex]\(-2\)[/tex] inside the parentheses:
[tex]\[
-2 \cdot (-5 x) + (-2) \cdot 4 \geq -10
\][/tex]
This simplifies to:
[tex]\[
10 x - 8 \geq -10
\][/tex]
#### Line (2):
Next, we need to isolate [tex]\(x\)[/tex]. To do this, we add 8 to both sides of the inequality:
[tex]\[
10 x - 8 + 8 \geq -10 + 8
\][/tex]
Which simplifies to:
[tex]\[
10 x \geq -2
\][/tex]
Divide both sides by 10:
[tex]\[
x \geq -\frac{2}{10}
\][/tex]
Which simplifies to:
[tex]\[
x \geq -\frac{1}{5}
\][/tex]
Let's compare with the calculations provided:
- The intermediate calculations ([tex]\(-5 x + 4 \geq 5\)[/tex] and [tex]\(-5 x \geq 1\)[/tex]) are incorrect.
### Correct Steps:
1. Start with [tex]\(-2(-5x + 4) \geq -10\)[/tex]
2. Distribute: [tex]\(10x - 8 \geq -10\)[/tex]
3. Add 8 to both sides: [tex]\(10x \geq -2\)[/tex]
4. Divide by 10: [tex]\(x \geq -\frac{1}{5}\)[/tex]
Therefore, Line (2) and Line (3) contain the errors. The correct intermediate step directly derived from Line (1) should have led to:
Line (2): [tex]\(10x - 8 \geq -10\)[/tex],
Line (3): [tex]\(10x \geq -2\)[/tex].
So, the correct final inequality, [tex]\(\boxed{x \geq -\frac{1}{5}}\)[/tex], confirms that the prior steps had calculation errors. The final correct step is [tex]\(x \geq -\frac{1}{5}\)[/tex] as derived from the initial inequality.
