Solve the inequation `5(x-2) gt 4(x+3) -24` and represent its solution on a number line. Given the replacement set is {-4,-3,-2,-1,0,1,2,3,4}.

Answer

To solve the inequation \(5(x - 2) > 4(x + 3) - 24\) and represent its solution on a number line, we can follow these steps:

### Step 1: Expand both sides of the inequation
Start by distributing the terms on both sides of the inequation.

\[ 5(x - 2) > 4(x + 3) - 24 \]

Expanding both sides gives:

\[ 5x - 10 > 4x + 12 - 24 \]

### Step 2: Simplify the right side
Now simplify the right side of the inequation:

\[ 5x - 10 > 4x + 12 - 24 \]
\[ 5x - 10 > 4x - 12 \]

### Step 3: Isolate the variable \(x\)
Next, we want to get all terms involving \(x\) on one side and constant terms on the other side. Subtract \(4x\) from both sides:

\[ 5x - 4x - 10 > -12 \]
\[ x - 10 > -12 \]

Now, add \(10\) to both sides to isolate \(x\):

\[ x > -12 + 10 \]
\[ x > -2 \]

### Step 4: Identify the solution set from the replacement set
The solution to the inequation is \(x > -2\). Now we need to find which values from the replacement set \(\{-4, -3, -2, -1, 0, 1, 2, 3, 4\}\) satisfy this condition.

The values greater than \(-2\) from the replacement set are:
\[ -1, 0, 1, 2, 3, 4 \]

### Step 5: Represent the solution on a number line
To represent the solution set on a number line, we will plot the values \(-1, 0, 1, 2, 3, 4\) with closed circles to indicate that these values are included in the solution set.

1. Draw a horizontal line to represent the number line.
2. Mark points for \(-4, -3, -2, -1, 0, 1, 2, 3, 4\).
3. Place closed circles on \(-1, 0, 1, 2, 3, 4\) to indicate these values are part of the solution.

### Final Solution
The solution set is \(\{-1, 0, 1, 2, 3, 4\}\).

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