Let's solve the inequality step-by-step:
Given inequality: [tex]\(4x - 9 > 2x + 14\)[/tex]
### Step 1: Isolate terms involving [tex]\(x\)[/tex]
First, subtract [tex]\(2x\)[/tex] from both sides of the inequality to gather [tex]\(x\)[/tex] terms on one side:
[tex]\[ 4x - 9 - 2x > 2x + 14 - 2x \][/tex]
This simplifies to:
[tex]\[ 2x - 9 > 14 \][/tex]
### Step 2: Isolate the constant
Next, add 9 to both sides of the inequality to gather constants on the other side:
[tex]\[ 2x - 9 + 9 > 14 + 9 \][/tex]
This simplifies to:
[tex]\[ 2x > 23 \][/tex]
### Step 3: Solve for [tex]\(x\)[/tex]
Now, divide both sides of the inequality by 2 to solve for [tex]\(x\)[/tex]:
[tex]\[ \frac{2x}{2} > \frac{23}{2} \][/tex]
This simplifies to:
[tex]\[ x > 11.5 \][/tex]
### Solution Set
The solution set for the inequality is all [tex]\(x\)[/tex] values greater than 11.5.
### Graphing the Solution Set
When graphing this solution set on a number line, you would:
1. Use an open circle (or dot) at [tex]\(x = 11.5 \)[/tex] to indicate that 11.5 is not included in the solution set.
2. Shade the region to the right of 11.5 to represent all values greater than 11.5.
So the final answer is:
[tex]\[ x > 11.5 \][/tex]
