Solve the inequality. Graph the solution set and write it in interval notation.
−3(x−4)−6x
<
−(4x+2)+2x
Question content area bottom
Part 1
Choose the correct graph below.
A.
An infinite number line has a tick mark labeled 0. The region to the right of a left bracket labeled 2 is shaded.
02
B.
An infinite number line has a tick mark labeled 0. The region to the right of a left parenthesis labeled 2 is shaded.
02
Your answer is correct.
C.
An infinite number line has a tick mark labeled 0. The region to the left of a right bracket labeled 2 is shaded.
02
D.
An infinite number line has a tick mark labeled 0. The region to the left of a right parenthesis labeled 2 is shaded.
02
Part 2
Write the answer in interval notation.
The correct answer is:
A. An infinite number line has a tick mark labeled 0. The region to the right of a left bracket labeled 2 is shaded (02).
The solution set in interval notation is:
(2, ∞)
The perimeter of a rectangle is to be no greater than
60 centimeters and the length must be
25 centimeters. Find the maximum width of the rectangle.
x cm25 cm
Question content area bottom
Part 1
the perimeter
of the rectangle
is less than or equal to
60
↓
↓
↓
x+25+
enter your response here
▼
60
First, understand the problem. Then translate the statement into an inequality.
The perimeter of a rectangle is given by the formula P = 2(length + width).
In this case, the length is fixed at 25 cm.
We want the perimeter to be no greater than 60 cm.
So, the inequality representing this situation would be:
2(25 + width) ≤ 60
Simplifying, we get:
50 + 2width ≤ 60
Subtracting 50 from both sides:
2width ≤ 10
Dividing both sides by 2:
width ≤ 5
Therefore, the maximum width of the rectangle can be 5 cm.
To solve the inequality −3(x−4)−6x < −(4x+2)+2x, we need to simplify both sides of the inequality and solve for x.
First, distribute the -3 on the left side: -3*x + 3*4 - 6x < -4x - 2 + 2x
Simplifying further: -3x + 12 - 6x < -4x - 2 + 2x
Combine like terms: -9x + 12 < -2x - 2
Move all the x terms to one side and the constant terms to the other side: -9x + 2x < -2 - 12
Simplifying further: -7x < -14
Divide both sides by -7, remembering to reverse the inequality when dividing by a negative number: x > 2
Now let's graph the solution set and write it in interval notation.
To graph the solution set, we'll use an open circle at x = 2, since the inequality is strict (x > 2). Then, we'll shade the region to the right of 2 on the number line, representing all x values greater than 2.
The correct graph is:
A. An infinite number line has a tick mark labeled 0. The region to the right of a left bracket labeled 2 is shaded.
For interval notation, we write the solution set as (2, ∞), which means all x values greater than 2.
Hence, the final answer in interval notation is (2, ∞).