To graph the solution set for the inequality [tex]\(3x - 12 \geq 7x + 4\)[/tex], we need to follow a series of algebraic steps to find the range of [tex]\(x\)[/tex] values that satisfy the inequality.
Step 1: Simplify the Inequality
First, we'll move all terms involving [tex]\(x\)[/tex] to one side of the inequality and the constant terms to the other side.
Starting with:
[tex]\[ 3x - 12 \geq 7x + 4 \][/tex]
Subtract [tex]\(7x\)[/tex] from both sides:
[tex]\[ 3x - 7x - 12 \geq 4 \][/tex]
[tex]\[ -4x - 12 \geq 4 \][/tex]
Next, add 12 to both sides to isolate the term involving [tex]\(x\)[/tex]:
[tex]\[ -4x - 12 + 12 \geq 4 + 12 \][/tex]
[tex]\[ -4x \geq 16 \][/tex]
Step 2: Solve for [tex]\(x\)[/tex]
To solve for [tex]\(x\)[/tex], we need to divide both sides by [tex]\(-4\)[/tex]. Remember that dividing both sides of an inequality by a negative number reverses the direction of the inequality:
[tex]\[ -4x \geq 16 \][/tex]
[tex]\[ x \leq -\frac{16}{4} \][/tex]
[tex]\[ x \leq -4 \][/tex]
Step 3: Graph the Solution Set
Now that we have [tex]\( x \leq -4 \)[/tex], we can graph this on the number line and the coordinate plane.
1. Number Line Representation:
- Draw a number line.
- Place a solid dot at [tex]\( -4 \)[/tex] because the inequality includes [tex]\( -4 \)[/tex] ([tex]\( \leq \)[/tex]).
- Shade the number line to the left of [tex]\( -4 \)[/tex] to represent all values [tex]\( x \leq -4 \)[/tex].
2. Coordinate Plane Representation:
- Draw the vertical line [tex]\( x = -4 \)[/tex].
- Shade the region to the left of the line to indicate all points where [tex]\( x \leq -4 \)[/tex].
- The vertical line [tex]\( x = -4 \)[/tex] should be solid to indicate that [tex]\( x = -4 \)[/tex] is included in the solution set.
3. Comparison of Functions:
- For further clarity, you can plot the functions [tex]\( y = 3x - 12 \)[/tex] and [tex]\( y = 7x + 4 \)[/tex] on the same coordinate plane.
- Identify the points where these two lines intersect [tex]\(x = -4\)[/tex].
- Shade the region to the left of the vertical line where [tex]\(y = 3x - 12\)[/tex] is greater than or equal to [tex]\(y = 7x + 4\)[/tex].
Step 4: Verify
To ensure the solution is correct, we can substitute [tex]\( x = -4 \)[/tex] into the original inequality and verify:
[tex]\[ 3(-4) - 12 \geq 7(-4) + 4 \][/tex]
[tex]\[ -12 - 12 \geq -28 + 4 \][/tex]
[tex]\[ -24 \geq -24 \][/tex]
This holds true, confirming that our solution [tex]\( x \leq -4 \)[/tex] is correct.
By following these steps, we have successfully graphed and verified the solution set for the inequality [tex]\(3x - 12 \geq 7x + 4\)[/tex].
