A - The solution to the inequality -6(x - 3) > 42 is x < -4. B - Graphically, this is represented by shading the region to the left of -4 on the number line with an open circle at -4. C - The inequality -6x + 18 > 42 has the solution x < -4.
A - Distribute the -6 on the left side: -6x + 18 > 42
Subtract 18 from both sides: -6x > 24
Divide both sides by -6 (since you're dividing by a negative number, the inequality sign will flip): x < -4
So, the solution to the inequality is x < -4.
B - To graph the solution x < -4 on a number line:
Locate -4 on the number line.
Shade the region to the left of -4 because the inequality is x < -4.
Use an open circle at -4 (since the inequality is strict, not including -4).
The graph visually represents all the values of x that satisfy the given inequality.
C - Simplifying the inequality:
We can start by simplifying the left side of the inequality:
-6(x - 3) = -6x + 18
This gives us the new inequality: -6x + 18 > 42
Isolating x:
To solve for x, we need to isolate it on one side.
Subtracting 18 from both sides, we get:
-6x > 24
Dividing by the coefficient of x:
To get x by itself, we need to divide both sides by -6. However, remember that dividing by a negative number flips the direction of the inequality.
x < -4
Therefore, any value of x less than -4 will make the inequality true.
While any number less than -4 works, let's look at -5 and -6 as examples:
For x = -5: -6(-5 - 3) = -30, which is greater than 42.
For x = -6: -6(-6 - 3) = 36, which is also greater than 42.