Question

Find the error in the calculations below, if there is one:

Line (1)
: -5_<2x+7<9

Line (2)
: -12_<2x<2

Line (3)
: -6_<x<1

Line (4)
:

An image shows a number line. There is a filled-in circle at negative 6 and an open circle at positive 1. The number line is shaded between negative 6 and 1.

Answer 1

Let's analyze the inequalities step by step to find any possible errors.

Starting with Line (1):
\[
-5 \leq 2x + 7 < 9
\]

We can break this compound inequality into two parts.

### Part 1:
\[
-5 \leq 2x + 7
\]
Subtract 7 from both sides:
\[
-5 - 7 \leq 2x \Rightarrow -12 \leq 2x
\]
Dividing both sides by 2:
\[
-6 \leq x \quad \text{(or rewritten as } x \geq -6)
\]

### Part 2:
\[
2x + 7 < 9
\]
Subtract 7 from both sides:
\[
2x < 9 - 7 \Rightarrow 2x < 2
\]
Dividing both sides by 2:
\[
x < 1
\]

Now we can combine the results from both parts.

From Part 1, we have:
\[
-6 \leq x \quad \text{(or } x \geq -6)
\]
From Part 2, we have:
\[
x < 1
\]

Putting both parts together, we get:
\[
-6 \leq x < 1
\]

This matches Line (3):
\[
-6 \leq x < 1
\]

There are no errors in the statements or the final result:
The number line has a filled-in circle at \(-6\) (indicating it is included) and an open circle at \(1\) (indicating it is not included). The shading from \(-6\) to \(1\) is correct.

In conclusion, none of the lines contain errors and thus all calculations appear to be correct.