Question

Justify each step by identifying the property used. Like this one 1/2(y+3)>1/3(4-y)

Answer 1

Answer:

$y>-\dfrac{1}{5}$

Step-by-step explanation:

Given:

$\dfrac{1}{2}(y+3)>\dfrac{1}{3}(4-y)$

Step 1: Use distributive property:

$\dfrac{1}{2}y+3\cdot \dfrac{1}{2}>\dfrac{1}{3}\cdot 4-\dfrac{1}{3}y$

Step 2: Using addition property, add $\dfrac{1}{3}y$ to both sides of inequality:

$\dfrac{1}{2}y+\dfrac{1}{3}y+\dfrac{3}{2}>\dfrac{4}{3}-\dfrac{1}{3}y+\dfrac{1}{3}y\\ \\\dfrac{1}{2}y+\dfrac{1}{3}y+\dfrac{3}{2}>\dfrac{4}{3}$

Step 3: Using addition property, add $-\dfrac{3}{2}$ to both sides of inequality:

$\dfrac{1}{2}y+\dfrac{1}{3}y+\dfrac{3}{2}-\dfrac{3}{2}>\dfrac{4}{3}-\dfrac{3}{2}\\ \\\dfrac{1}{2}y+\dfrac{1}{3}y>\dfrac{4}{3}-\dfrac{3}{2}$

Step 4: Subtract fractions in the right side:

$\dfrac{4}{3}-\dfrac{3}{2}=\dfrac{4\cdot 2-3\cdot 3}{3\cdot 3}=-\dfrac{1}{6}$

Step 5: Use distributive property in left side:

$\dfrac{1}{2}y+\dfrac{1}{3}y=y\left(\dfrac{1}{2}+\dfrac{1}{3}\right)=\dfrac{3+2}{2\cdot 3}y=\dfrac{5}{6}y$

You get the inequality:

$\dfrac{5}{6}y>-\dfrac{1}{6}$

Step 6: Use division property:

$y>-\dfrac{1}{6}:\dfrac{5}{6}\\ \\y>-\dfrac{1}{6}\cdot \dfrac{6}{5}\\ \\y>-\dfrac{1}{5}$