1. Negative Absolute Value x+4 ABV<-4
2. 3 ABV x-6 ABV greater than or equal to -2
Don't know what you have against symbols, but if I interpret your words correctly, we have
-|x+4| < -4
|x+4| > 4
Now, since
|n| = n if n>=0
|n| = -n if n<0, we have two cases:
x+4 > 0 (that is, x > -4)
x+4 > 4
x > 0
x+4 < 0 (that is, x < -4)
-(x+4) > 4
x+4 < -4
x < -8
So, the solution set is x < -8 or x > 0
Think about the shape of the graph.
f(x) = -|x+4| is an upside-down V shape, so if you want it to be less than -4, draw a line at y = -4, and you will see that the region where the graph is below the line is what we calculated above.
Work the other the same way:
3|x-6| >= -2
Is this correct?
x>=16/3
x<=20/3
To solve these absolute value inequalities, we need to isolate the absolute value expressions and consider their positive and negative cases separately. Let's solve each inequality step by step:
1. Negative Absolute Value (x + 4) ABV < -4:
a) Considering the positive case:
x + 4 < -4
Solving for x:
x < -4 -4
x < -8
b) Considering the negative case:
-(x + 4) < -4
Multiplying both sides by -1 (which reverses the inequality direction):
x + 4 > 4
Solving for x:
x > 4 - 4
x > 0
Therefore, the solution to the inequality is x < -8 or x > 0.
2. 3 ABV x - 6 ABV ≥ -2:
a) Considering the positive case:
3(x) - 6(x) ≥ -2
Simplifying:
3x - 6x ≥ -2
-3x ≥ -2
Dividing both sides by -3 (which reverses the inequality direction since we are dividing by a negative number):
x ≤ -2/-3
x ≤ 2/3
b) Considering the negative case:
-3(x) - 6(-x) ≥ -2
Simplifying:
-3x + 6x ≥ -2
3x ≥ -2
Dividing both sides by 3:
x ≥ -2/3
Therefore, the solution to the inequality is x ≤ 2/3 or x ≥ -2/3.
Remember to always consider both the positive and negative cases when working with absolute value inequalities to ensure you capture all possible solutions.