Question

Which graph below bests represent the solution set for `-2x+5+x\le3x+13`?.....

Answer 1

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



Which number line is the correct representation of the solution?

Answer 2

To solve the inequality \(-2x + 5 + x \leq 3x + 13\), we first combine like terms:

1. Combine terms on the left side:
\[
-2x + x + 5 \leq 3x + 13
\]
This simplifies to:
\[
-x + 5 \leq 3x + 13
\]

2. Now, we will isolate \(x\). First, we can add \(x\) to both sides:
\[
5 \leq 4x + 13
\]

3. Next, subtract \(13\) from both sides:
\[
5 - 13 \leq 4x
\]
This gives us:
\[
-8 \leq 4x
\]

4. Now divide both sides by \(4\):
\[
-\frac{8}{4} \leq x
\]
Which simplifies to:
\[
-2 \leq x
\]
or equivalently:
\[
x \geq -2
\]

The solution set for the inequality is \(x \geq -2\), which can be represented graphically as a number line where all numbers greater than or equal to \(-2\) are included.

In a graphing representation, this would look like a filled (or solid) point at \(-2\) with a line extending to the right towards positive infinity. Look for a graph that represents that.

Answer 3

To solve the compound inequality \(-7 \leq 2x + 1 < 11\), we will break it down into two parts and solve each part step-by-step.

### Step 1: Solve the left side of the compound inequality

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

Subtract \(1\) from both sides:

\[
-7 - 1 \leq 2x
\]
\[
-8 \leq 2x
\]

Now, divide both sides by \(2\):

\[
-4 \leq x
\]
or equivalently,
\[
x \geq -4
\]

### Step 2: Solve the right side of the compound inequality

\[
2x + 1 < 11
\]

Subtract \(1\) from both sides:

\[
2x < 11 - 1
\]
\[
2x < 10
\]

Now, divide both sides by \(2\):

\[
x < 5
\]

### Step 3: Combine the two parts

Putting it all together, we have:

\[
-4 \leq x < 5
\]

### Final Result

The solution for the given compound inequality is:

\[
-4 \leq x < 5
\]

### Representation on a Number Line

On a number line, this solution can be represented as:

- A **closed circle** (or solid point) on \(-4\) (to indicate that \(-4\) is included in the solution).
- An **open circle** (or hollow point) on \(5\) (to indicate that \(5\) is not included in the solution).
- A line extending from \(-4\) to \(5\).

Look for a number line graph that has these characteristics:

- Closed (filled) dot at \(-4\)
- Open (unfilled) dot at \(5\)
- A line connecting them, indicating that all numbers in this interval are included in the solution.