Sure, let's solve the inequality \( 5 - 2x > x + 2 \) step-by-step.
### Step 1: Move all terms involving \( x \) to one side of the inequality and constants to the other side.
Start with the inequality:
[tex]\[ 5 - 2x > x + 2 \][/tex]
Subtract \( x \) from both sides to start isolating \( x \):
[tex]\[ 5 - 2x - x > 2 \][/tex]
Simplify:
[tex]\[ 5 - 3x > 2 \][/tex]
### Step 2: Isolate the variable \( x \).
Subtract 5 from both sides to move the constant terms to the right side:
[tex]\[ 5 - 5 - 3x > 2 - 5 \][/tex]
Simplify:
[tex]\[ -3x > -3 \][/tex]
### Step 3: Solve for \( x \).
To isolate \( x \), divide both sides of the inequality by \(-3\). Recall that dividing by a negative number reverses the inequality sign:
[tex]\[ x < 1 \][/tex]
So the solution to the inequality is:
[tex]\[ x < 1 \][/tex]
### Step 4: Illustrate the result on the number line.
To illustrate the solution on a number line:
- Draw a number line.
- Mark a point at \( x = 1 \).
- Since \( x \) is less than 1 (but not equal to 1), use an open circle at \( x = 1 \).
- Shade the number line to the left of \( x = 1 \), indicating all real numbers less than 1.
The solution set in interval notation is:
[tex]\[ (-\infty, 1) \][/tex]
This means that any real number less than 1 is a solution to the inequality [tex]\( 5 - 2x > x + 2 \)[/tex].
