Sove the inequality x+2/4-x+1/3>1/2
x - x = 0
Now you see why parentheses are important:
(x+2)/(4-x) + 1/3 > 1/2
(x+2)/(4-x) > 1/6
If x<4,
6(x+2) > 4-x
6x+12 > 4-x
7x > -8
x > -8/7
So, one solution is the interval (-8/7,4)
If x>4,
6(x+2) < 4-x
x < -8/7
Not possible, sine we started out with x>4
To solve the inequality x + 2/(4 - x) + 1/3 > 1/2, we need to manipulate the expression to isolate x on one side of the inequality sign. Here's how you can do it step by step:
Step 1: Simplify the expression on the left side of the inequality:
x + 2/(4 - x) + 1/3 > 1/2
To add fractions, we need a common denominator which is 6 in this case. Multiply the first fraction by 3/3 and the second fraction by 2/2:
x + (2 * 3) / (3 * (4 - x)) + (1 * 2) / (2 * 3) > 1/2
Simplify the expression further:
x + 6 / (12 - 3x) + 2 / 6 > 1/2
Step 2: Take care of the fractions by finding a common denominator. Multiply each term by 6 to eliminate the fractions:
6x + 36 / (12 - 3x) + 4 > 3
Simplify the expression further:
6x + 36 + 4(12 - 3x) > 3(12 - 3x)
Distribute the 4 on the left side and the 3 on the right side:
6x + 36 + 48 - 12x > 36 - 9x
Step 3: Combine like terms on each side of the inequality:
-6x + 84 > 36 - 9x
Step 4: Get all the x terms on one side and all the constants on the other side of the inequality sign:
-6x + 9x > 36 - 84
Simplify:
3x > -48
Step 5: Divide both sides of the inequality by 3 to solve for x:
x > -48/3
Simplify:
x > -16
The solution to the inequality x + 2/(4 - x) + 1/3 > 1/2 is x > -16.