To solve the inequality [tex]\( 3x - 12 \geq 7x + 4 \)[/tex] and graph the solution set, follow these steps:
1. Rearrange the Inequality:
Start with the given inequality:
[tex]\[
3x - 12 \geq 7x + 4
\][/tex]
2. Combine Like Terms:
Isolate all terms containing [tex]\( x \)[/tex] on one side by subtracting [tex]\( 7x \)[/tex] from both sides:
[tex]\[
3x - 7x - 12 \geq 4
\][/tex]
Simplify:
[tex]\[
-4x - 12 \geq 4
\][/tex]
3. Isolate x:
Add 12 to both sides to further isolate the term with [tex]\( x \)[/tex]:
[tex]\[
-4x - 12 + 12 \geq 4 + 12
\][/tex]
Simplify:
[tex]\[
-4x \geq 16
\][/tex]
4. Solve for x:
Divide both sides by -4. Remember, when dividing or multiplying both sides of an inequality by a negative number, the inequality sign reverses:
[tex]\[
x \leq -4
\][/tex]
5. Write the Solution Set:
The solution to the inequality is all [tex]\( x \)[/tex] that satisfy [tex]\( x \leq -4 \)[/tex]. In interval notation, this is written as:
[tex]\[
(-\infty, -4]
\][/tex]
6. Graph the Solution Set:
On a number line, you will shade all the numbers to the left of [tex]\(-4\)[/tex], including [tex]\(-4\)[/tex]. Use a closed circle to indicate that [tex]\(-4\)[/tex] is part of the solution set.
Here's the graph on the number line:
-∞ <-----------------[===================> -4
The closed circle at [tex]\(-4\)[/tex] signifies that [tex]\( x = -4 \)[/tex] is included in the solution set, and the arrow extending to the left indicates that all values less than [tex]\(-4\)[/tex] are also included.
