Question

Solve the inequality both algebraically and graphically. Draw a number line graph of the solution and give interval notation.
4xplus3less thanminusStartFraction 3 x Over 2 EndFraction
plus5

Answer 1

To solve the inequality \( 4x + 3 < -\frac{3x}{2} + 5 \), we will start by isolating \( x \).

### Step 1: Combine like terms

First, let's eliminate the fraction by multiplying both sides by \( 2 \) to make the computations easier:

\[
2(4x + 3) < 2\left(-\frac{3x}{2} + 5\right)
\]

This simplifies to:

\[
8x + 6 < -3x + 10
\]

### Step 2: Move all \( x \) terms to one side

Now, we will add \( 3x \) to both sides:

\[
8x + 3x + 6 < 10
\]

This simplifies to:

\[
11x + 6 < 10
\]

### Step 3: Isolate \( x \)

Next, we subtract \( 6 \) from both sides:

\[
11x < 4
\]

Then, we divide by \( 11 \):

\[
x < \frac{4}{11}
\]

### Solution in Interval Notation

The solution to the inequality is:

\[
(-\infty, \frac{4}{11})
\]

### Step 4: Graphical Solution

On a number line, we would represent this interval with an open circle at \( \frac{4}{11} \) to indicate that \( \frac{4}{11} \) is not included in the solution. The line would extend leftwards towards negative infinity.

Here's a representation of the number line:

```
<=====o====================>
-∞ 4/11
```

**Summary:**
- The solution set is \( x < \frac{4}{11} \)
- In interval notation, the solution is \( (-\infty, \frac{4}{11}) \)