Question

Solve: `-7\le2x+1<11`



Which number line is the correct representation of the solution?

.....

Answer 1

12x - 3 less than equal too 15x or -0.2x + 10 < 5

Which choice best describes the solution set?

1\le x<-25

-1\le x<25

x\ge1 or <-25

x\le-1 or x>25

Answer 2

To solve the compound inequality \(-7 \leq 2x + 1 < 11\), we can break it down into two parts:

1. \(-7 \leq 2x + 1\)
2. \(2x + 1 < 11\)

Let's solve each part separately.

### Part 1: \(-7 \leq 2x + 1\)

Subtract 1 from both sides:

\[
-7 - 1 \leq 2x
\]

\[
-8 \leq 2x
\]

Now, divide both sides by 2 (note that we do not flip the inequality here since we are dividing by a positive number):

\[
-4 \leq x
\]

This can also be written as:

\[
x \geq -4
\]

### Part 2: \(2x + 1 < 11\)

Subtract 1 from both sides:

\[
2x < 11 - 1
\]

\[
2x < 10
\]

Now, divide both sides by 2:

\[
x < 5
\]

### Combining the Results

Now we can combine our results from both parts:

\[
-4 \leq x < 5
\]

### Number Line Representation

To represent the solution \(-4 \leq x < 5\) on a number line:

- Place a closed circle on -4 (indicating that it is included).
- Place an open circle on 5 (indicating that it is not included).
- Shade the region between -4 and 5.

This gives us the correct representation of the solution on the number line.

Answer 3

To solve the inequalities you've provided, let's analyze them one at a time.

### First Inequality: \( 12x - 3 \leq 15x \)

1. Start by isolating the variable \(x\):
\[
12x - 3 \leq 15x
\]
Subtract \(12x\) from both sides:
\[
-3 \leq 3x
\]
Now, divide both sides by 3:
\[
-1 \leq x
\]
This can be rewritten as:
\[
x \geq -1
\]

### Second Inequality: \( -0.2x + 10 < 5 \)

1. Start by isolating the variable \(x\):
\[
-0.2x + 10 < 5
\]
Subtract 10 from both sides:
\[
-0.2x < 5 - 10
\]
Which simplifies to:
\[
-0.2x < -5
\]
Now, divide both sides by -0.2. Remember to flip the inequality sign when dividing by a negative number:
\[
x > \frac{-5}{-0.2} = 25
\]
Thus, we have:
\[
x > 25
\]

### Combining the Results

Now we need to combine the results of the two inequalities:

1. The first inequality gives us \(x \geq -1\).
2. The second inequality gives us \(x > 25\).

The solution set includes both conditions:

\[
x \geq -1 \quad \text{or} \quad x > 25
\]

### Evaluating the Choices

Now let's evaluate the choices based on what we've derived:

1. \(1 \le x < -25\) - Incorrect
2. \(-1 \le x < 25\) - Incorrect
3. \(x \ge 1\) or \(x < -25\) - Incorrect
4. \(x \le -1\) or \(x > 25\) - Incorrect

From our solution, the correct format would be:

\[
x \ge -1 \quad \text{or} \quad x > 25
\]

Since none of the provided choices perfectly match our derived answer, there might be an oversight in the options listed. Based on the solutions:

- The correct answer from the given conditions could be summarized as \(x \ge -1\) or \(x > 25\), which isn't listed. Assuming the options are correctly specified, select the option that best aligns with individual inequalities.

If you have an option that states \(x \ge -1\) or \(x > 25\) that should be chosen.