Solve |2y+6|<10 for inequality and write the solution in interval notation and graph
(2y+6)<10 and -(2y+6)<10
2 y < 4 and -2y - 6 < 10
y < 2 and -y < 8 (which is y > -8)
looks like between -8 and +2
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check
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well try y = 0, yes |6| is <10
try y = -7, yes |-8| is <10
try y = +1, yes |+8| is <10
|2y+6|<10
2y+5 < 10 AND -2y-6 < 10
2y < 5 AND -2y < 16
y < 5/2 AND y > -8
in traditional notation: -8 < y < 5/2
I will let you express it in interval notation
your graph should consist of a line joining -8 to 5/2, with the end points
consisting of open circles.
that is, you want all the values between -8 and 5/2
argghhh, don't know how my 6 turned into a 5 in line 2
go with "anonymous"
|2y+6|<10
|y+3| < 5
(y+3)^2 < 5^2
y^2+6y+9 < 25
y^2+6y-16 < 0
(y-2)(y+8) < 0
solution interval: (-8,2)
graph is at
https://www.wolframalpha.com/input/?i=%7C2y%2B6%7C%3C10
or,
|2y+6| < 10
|y+3| < 5
so you want all the points within 5 of the value -3.
-3-5 < y < -3+5
-8 < y < 2
To solve the inequality |2y + 6| < 10, we need to break it down into two separate inequalities and solve for each one. Let's start by removing the absolute value bars.
1. When 2y + 6 is positive (or zero):
For the equation 2y + 6 < 10, we can subtract 6 from both sides:
2y + 6 - 6 < 10 - 6
2y < 4
Next, divide both sides by 2 to solve for y:
y < 2
2. When 2y + 6 is negative:
For the equation -(2y + 6) < 10, we need to distribute the negative sign:
-2y - 6 < 10
Add 6 to both sides:
-2y < 16
Divide both sides by -2, remembering to reverse the inequality symbol because we are dividing by a negative number:
y > -8
So, we have two inequalities:
1. y < 2
2. y > -8
To express the solution in interval notation, we can write it as the union of these two intervals. Interval notation uses square brackets [ ] for closed intervals and parentheses ( ) for open intervals.
The solution in interval notation is (-∞, -8) ∪ (2, ∞).
To graph the solution on a number line, draw a line with a marked point at -8 and a marked point at 2. Then, draw an open circle at -8 and an open circle at 2 to represent that these points are not included in the solution. Finally, shade the region between -8 and 2 to indicate the range of values that satisfy the inequality.
Here is a visual representation:
-------------------------------------------------------
(-∞).....................(-8).............(2).............(∞)
X X
The shaded area between -8 and 2 represents the solution to the inequality |2y + 6| < 10.