Right now we are learning about absolute value inequalities. I have to solve and graph this inequality:
Pretend my paranpheses are absolute value signs
(x-4) + 5 is greater than or equal to 10.
I assume I would take 5 away from each side first. But what would be the correct way to display this on a line, and what would be its set builder notation and interval notation?
Thank you so much! :-)
I took away 5 from each side and added 4 to each side. That got me
(x) is greater than or equal to 14.
Would this make the set builder notation be:
{x such as x is greater than or equal to 4}
And the interval notation be:
(negative infinity, 14] U [14, infinity)?
Thanks! :-)
You have to consider two cases. I will leave out the = sign for convenience.
(1) If x - 4 > 0, then
x - 4 + 5 > 10
x > 9 (which satisfies the x>4 assumption)
(2) If x - 4 < 0, then
If 4 - x + 5 > 10, then
-x > 1
x < -1
(which satisfies the x<4 assumption)
To solve the absolute value inequality (x-4) + 5 >= 10, you can start by subtracting 5 from both sides:
(x-4) >= 5
To solve this absolute value inequality, you need to consider two cases - one when the expression inside the absolute value is positive, and the other when it is negative.
When (x-4) >= 0 (positive):
In this case, the inequality simplifies to:
x-4 >= 5
Adding 4 to both sides gives:
x >= 9
When (x-4) < 0 (negative):
In this case, we need to change the inequality sign when removing the absolute value, so:
-(x-4) >= 5
Expanding the negative sign gives:
- x + 4 >= 5
Subtracting 4 from both sides gives:
- x >= 1
Now, since the inequality is multiplied by -1, we need to flip the inequality sign:
x <= -1
So, the solution to the absolute value inequality (x-4) + 5 >= 10 is:
x >= 9 or x <= -1
To graph this on a number line, you can represent the values of x between -∞ and -1, and the values of x from 9 to +∞ as shaded regions on the number line.
In set builder notation, the solution can be written as:
{x | x ≥ 9 or x ≤ -1}
In interval notation, the solution can be written as:
(-∞, -1] ∪ [9, +∞)