Question 345330: Simplify each side first, then solve the inequality. Write your answer with interval notation.
1/3t - 1/2(5 - t) < 0
Answer by haileytucki(390)   (Show Source):
You can put this solution on YOUR website! (1)/(3)*t-(1)/(2)*(5-t)<0
Reorder the polynomial 5-t alphabetically from left to right, starting with the highest order term.
(1)/(3)*t-(1)/(2)*(-t+5)<0
Multiply all the factors separated by a * in (1)/(3)*t-(1)/(2)*(-t+5).
(t)/(3)-(-t+5)/(2)<0
Multiply -1 by each term inside the parentheses.
(t)/(3)+(t-5)/(2)<0
Multiply each term by a factor of 1 that will equate all the denominators. In this case, all terms need a denominator of 6. The (t)/(3) expression needs to be multiplied by ((2))/((2)) to make the denominator 6. The ((t-5))/(2) expression needs to be multiplied by ((3))/((3)) to make the denominator 6.
(t)/(3)*(2)/(2)+(t-5)/(2)*(3)/(3)<0
Multiply the expression by a factor of 1 to create the least common denominator (LCD) of 6.
(t(2))/(6)+(t-5)/(2)*(3)/(3)<0
Multiply the expression by a factor of 1 to create the least common denominator (LCD) of 6.
(t(2))/(6)+((t-5)(3))/(6)<0
The numerators of expressions that have equal denominators can be combined. In this case, (t(2))/(6) and (((t-5)(3)))/(6) have the same denominator of 6, so the numerators can be combined.
(t(2)+((t-5)(3)))/(6)<0
Simplify the numerator of the expression.
(5t-15)/(6)<0
Factor out the GCF of 5 from each term in the polynomial.
(5(t)+5(-3))/(6)<0
Factor out the GCF of 5 from 5t-15.
(5(t-3))/(6)<0
Multiply each term in the inequality by 6.
(5(t-3))/(6)*6<0*6
Simplify the left-hand side of the inequality by canceling the common factors.
5(t-3)<0*6
Multiply 0 by 6 to get 0.
5(t-3)<0
Divide each term in the inequality by 5.
(5(t-3))/(5)<(0)/(5)
Simplify the left-hand side of the inequality by canceling the common factors.
t-3<(0)/(5)
0 divided by any number or variable is 0.
t-3<0
Since -3 does not contain the variable to solve for, move it to the right-hand side of the inequality by adding 3 to both sides.
t<3
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