Solve the following inequation, write down the solution set and represent it on the real number line :
`-2+10le13x+10lt24+10x,x inZ`

Answer

To solve the given inequation, we will follow the steps systematically:

### Step 1: Write down the inequation
The given inequation is:
\[ -2 + 10x \leq 13x + 10 \leq 24 + 10x \]

### Step 2: Break it down into two parts
We can break this into two separate inequalities:
1. \(-2 + 10x \leq 13x + 10\)
2. \(13x + 10 \leq 24 + 10x\)

### Step 3: Solve the first inequality
Starting with the first inequality:
\[ -2 + 10x \leq 13x + 10 \]
Subtract \(10x\) from both sides:
\[ -2 \leq 3x + 10 \]
Now, subtract \(10\) from both sides:
\[ -12 \leq 3x \]
Finally, divide by \(3\):
\[ -4 \leq x \quad \text{or} \quad x \geq -4 \]

### Step 4: Solve the second inequality
Now, let's solve the second inequality:
\[ 13x + 10 \leq 24 + 10x \]
Subtract \(10x\) from both sides:
\[ 3x + 10 \leq 24 \]
Now, subtract \(10\) from both sides:
\[ 3x \leq 14 \]
Finally, divide by \(3\):
\[ x \leq \frac{14}{3} \quad \text{or} \quad x \leq 4.67 \]

### Step 5: Combine the results
From the two inequalities, we have:
\[ -4 \leq x \leq \frac{14}{3} \]
Since \(x\) must be an integer, we can list the integers that satisfy this condition:
\[ x = -4, -3, -2, -1, 0, 1, 2, 3, 4 \]

### Step 6: Write down the solution set
The solution set in integers is:
\[ \{ -4, -3, -2, -1, 0, 1, 2, 3, 4 \} \]

### Step 7: Represent it on the real number line
To represent this on the real number line, we will mark the points corresponding to the integers from \(-4\) to \(4\).

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