Solve the inequality. Graph the solution set and write it in interval notation.
1/3(x+3)<1/4(2x+3)
To solve the inequality 1/3(x+3) < 1/4(2x+3), we will first simplify the expression on both sides.
Starting with the left side:
1/3(x+3) = (1/3)(x) + (1/3)(3) = (1/3)x + 1
Now simplifying the right side:
1/4(2x+3) = (1/4)(2x) + (1/4)(3) = (1/2)x + (3/4)
So now we have the simplified inequality:
(1/3)x + 1 < (1/2)x + (3/4)
To solve for x, we will isolate the x term on one side of the inequality.
Subtracting (1/3)x from both sides:
1 < (1/2)x - (1/3)x + (3/4)
Simplifying the fractional terms on the right side:
1 < (3/6)x - (2/6)x + (9/12)
Combine like terms and simplify further:
1 < (1/6)x + (9/12)
Now, we can rewrite the fractional inequality as a whole number inequality by multiplying both sides of the equation by 12 to clear the fractions.
Multiplying both sides by 12:
12 < 12[(1/6)x + (9/12)]
Simplifying the right side:
12 < 2x + 9
Subtracting 9 from both sides:
12 - 9 < 2x + 9 - 9
3 < 2x
Finally, dividing by 2 on both sides to solve for x:
(3/2) < (2x)/2
3/2 < x
So the solution to the inequality is x > 3/2.
Graphically, this means the solution is all x-values greater than 3/2 on the number line.
In interval notation, the solution can be written as (3/2, ∞).