4-3| m+2 |>-14
The answer shows -8 < m < 4, but I am not getting it. Could someone please explain?
You have two cases. Either
m+2 >= 0 or m+2 < 0
Case 1: m+2 >= 0 (m >= -2)
4-3(m+2) > -14
-3(m+2) > -18
m+2 < 6
m < 4
So, this solution is -2 <= m < 4
Case 2: m+2 < 0 (m < -2)
4+3(m+2) > -14
3(m+2) > -18
m+2 > -6
m > -8
So, this solution is -8 < m <= -2
Combine the solutions to get
-8 < m < 4
As a mental check, consider things this way
4-3|m+2| > -14
-3|m+2| > -18
|m+2| < 6
You know the shape of |x| is a v, so you want the interval where the v is below the line y=6.
|x| < 6 is -6 < x < 6
so, shifting the graph to the left 2 units, we have
|x+2| < 6 is -8 < x < 4
How did you get 18 and -18 on right hand side? I was thinking 4-3 = 1 and then solve.
If you saw 4-3x = -14
would you say 1x = -18?
No, you'd isolate all the x's first on one side.
Now 4-3+x = -14 would work the way you said.
To solve the inequality 4-3|m+2|>-14, let's break it down step by step:
Step 1: Remove the absolute value symbols.
We have |m+2|. When solving inequalities involving absolute values, we need to consider both a positive and a negative case. So we will rewrite the inequality with two cases: m+2 for the positive case, and -(m+2) for the negative case.
Positive case: 4 - 3(m+2) > -14
Negative case: 4 - 3(-(m+2)) > -14
Step 2: Simplify the expressions.
Positive case: 4 - 3m - 6 > -14
Negative case: 4 + 3m + 6 > -14
Simplifying further:
Positive case: -3m - 2 > -14
Negative case: 3m + 10 > -14
Step 3: Solve each case separately.
Let's first solve the positive case:
-3m - 2 > -14
We will now isolate m by moving -2 to the right side of the equation:
-3m > -14 + 2
-3m > -12
To solve for m, divide both sides by -3, remembering to reverse the inequality sign since we divide by a negative number:
m < -12/-3
m < 4
Now, let's solve the negative case:
3m + 10 > -14
We will now isolate m by moving 10 to the right side of the equation:
3m > -14 - 10
3m > -24
To solve for m, divide both sides by 3:
m > -24/3
m > -8
Step 4: Combine the solutions.
Since we have two cases, we need to consider where both cases are true. In this case, we are looking for the values of m that satisfy both m < 4 (positive case) and m > -8 (negative case). In other words, we need to find where -8 < m < 4.
Therefore, the solution to the inequality 4-3|m+2|>-14 is -8 < m < 4.