Solve the inequality. Graph the solution set and write it in interval notation.
negative 6 left parenthesis x minus 5 right parenthesis minus 4 x
less than
negative left parenthesis 5 x plus 6 right parenthesis plus 4 x
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Part 1
Choose the correct graph below.
A.
An infinite number line has a tick mark labeled 0. The region to the left of a right parenthesis labeled 4 is shaded.
0
4
B.
An infinite number line has a tick mark labeled 0. The region to the right of a left parenthesis labeled 4 is shaded.
0
4
C.
An infinite number line has a tick mark labeled 0. The region to the left of a right bracket labeled 4 is shaded.
0
4
D.
An infinite number line has a tick mark labeled 0. The region to the right of a left bracket labeled 4 is shaded.
0
4
To solve the inequality and graph the solution set, we first need to simplify the inequality.
Given: -6(x - 5) - 4x < -(5x + 6) + 4x
First, distribute the -6:
-6x + 30 - 4x < -(5x + 6) + 4x
Next, simplify the right side:
-6x + 30 - 4x < -5x - 6 + 4x
Combine like terms:
-10x + 30 < -x - 6
Now, let's isolate the x variable. Subtract -x and 30 from both sides:
-9x < -36
To solve for x, divide by -9 (and be sure to reverse the inequality because we are dividing by a negative number):
x > (-36) / (-9)
x > 4
So, x is greater than 4.
To graph the solution set, we draw a number line and mark a closed circle at 4 (since x is not equal to 4), and shade the region to the right of 4, indicating that the values of x that satisfy the inequality are greater than 4.
The correct graph is option B:
An infinite number line has a tick mark labeled 0. The region to the right of a left parenthesis labeled 4 is shaded.
Therefore, the solution set in interval notation is (4, ∞).