Given that `x in R`, solve the following inequality and graph the solution on the number the : `-1 le 3 + 4x lt 23`

Answer

To solve the inequality \(-1 \leq 3 + 4x < 23\), we will break it down into two parts and solve each part step by step.

### Step 1: Break down the compound inequality
The given compound inequality can be split into two separate inequalities:
1. \(-1 \leq 3 + 4x\)
2. \(3 + 4x < 23\)

### Step 2: Solve the first inequality
Starting with the first inequality:
\[ -1 \leq 3 + 4x \]
Subtract 3 from both sides:
\[ -1 - 3 \leq 4x \]
This simplifies to:
\[ -4 \leq 4x \]
Now, divide both sides by 4:
\[ -1 \leq x \]
This can also be written as:
\[ x \geq -1 \]

### Step 3: Solve the second inequality
Now, let's solve the second inequality:
\[ 3 + 4x < 23 \]
Subtract 3 from both sides:
\[ 4x < 23 - 3 \]
This simplifies to:
\[ 4x < 20 \]
Now, divide both sides by 4:
\[ x < 5 \]

### Step 4: Combine the results
Now we combine the results from both inequalities:
\[ -1 \leq x < 5 \]
This can be expressed as:
\[ -1 \leq x < 5 \]
or in interval notation:
\[ [-1, 5) \]

### Step 5: Graph the solution on the number line
To graph the solution on a number line:
- Draw a number line and mark the points -1 and 5.
- Since -1 is included in the solution (indicated by the "less than or equal to"), we will use a closed dot at -1.
- Since 5 is not included (indicated by the "less than"), we will use an open dot at 5.
- Shade the region between -1 and 5 to indicate all the values that satisfy the inequality.

### Final Answer
The solution set is:
\[ -1 \leq x < 5 \]
Graphically represented on the number line with a closed dot at -1 and an open dot at 5.

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