how do you solve this compound inequality
5>-4x+4 or 9<or equal to -3x+4
5>-4x+4
1>-4x
(1/4) > -x
-1/4 <x change direction of arrow when multiplying both sides by negative value.
so
x> -1/4 is in (everything right of -1/4)
now
9</=-3x+4
5 </= -3x
-5/3 >/= x
so
x </= -5/3 is in (-5/3 and everything left of that)
graph that
The only part of the x axis that is not in your domain is between right of -5/3 and -1/4 including -1/4
To solve a compound inequality, you need to solve each inequality separately and then find the intersection or union of their solutions, depending on the type of compound inequality.
Let's solve each inequality step by step.
Inequality 1: 5 > -4x + 4
1. Subtract 4 from both sides: 5 - 4 > -4x + 4 - 4
Simplify: 1 > -4x
2. Divide both sides by -4 (note the direction of the inequality changes when dividing by a negative number): (1/-4) < (-4x/-4)
Simplify: -1/4 < x or x > -1/4
Inequality 2: 9 ≤ -3x + 4
1. Subtract 4 from both sides: 9 - 4 ≤ -3x + 4 - 4
Simplify: 5 ≤ -3x
2. Divide both sides by -3: (5/-3) ≥ (-3x/-3)
Simplify: -5/3 ≥ x or x ≤ -5/3
Now, we have the two separate solutions:
-1/4 < x and x ≤ -5/3
To express the compound inequality, you need to identify whether it represents an intersection (AND) or a union (OR) of the individual solutions.
The original compound inequality is: 5 > -4x + 4 or 9 ≤ -3x + 4.
Since the connective between the two inequalities is "OR," we take the union of the individual solutions.
Final Solution:
x ∈ (-∞, -5/3] U (-1/4, ∞)