Sure, let's solve the inequality step-by-step.
The given inequality is:
[tex]\[ 2x - 6 \geq 6(x - 2) + 8 \][/tex]
First, let's expand and simplify the right side of the inequality:
[tex]\[ 6(x - 2) + 8 \][/tex]
[tex]\[ = 6x - 12 + 8 \][/tex]
[tex]\[ = 6x - 4 \][/tex]
So the inequality becomes:
[tex]\[ 2x - 6 \geq 6x - 4 \][/tex]
Next, we need to get all terms involving [tex]\( x \)[/tex] on one side and constant terms on the other side. Let's subtract [tex]\( 6x \)[/tex] from both sides:
[tex]\[ 2x - 6 - 6x \geq 6x - 4 - 6x \][/tex]
This simplifies to:
[tex]\[ -4x - 6 \geq -4 \][/tex]
Now, let's add 6 to both sides to isolate the term with [tex]\( x \)[/tex]:
[tex]\[ -4x - 6 + 6 \geq -4 + 6 \][/tex]
This simplifies to:
[tex]\[ -4x \geq 2 \][/tex]
To solve for [tex]\( x \)[/tex], we need to divide both sides by -4. Remember, when we divide or multiply an inequality by a negative number, we need to reverse the inequality sign:
[tex]\[ x \leq \frac{2}{-4} \][/tex]
[tex]\[ x \leq -\frac{1}{2} \][/tex]
Therefore, the solution to the inequality [tex]\( 2x - 6 \geq 6(x - 2) + 8 \)[/tex] is:
[tex]\[ x \leq -\frac{1}{2} \][/tex]
Now let's represent this solution on the number line.
- Draw a number line.
- Identify the point [tex]\( -\frac{1}{2} \)[/tex] on the number line.
- Shade all the numbers to the left of [tex]\( -\frac{1}{2} \)[/tex], including [tex]\( -\frac{1}{2} \)[/tex], since it is part of the solution.
This way, every number less than or equal to [tex]\( -\frac{1}{2} \)[/tex] will be part of the solution set for the inequality.
