$\displaystyle -7y + 10 > 3y + 12$

$\displaystyle -7y + 10 - 3y > 3y + 12 - 3y$

$\displaystyle -10y + 10 > 12$

$\displaystyle -10y + 10 - 10 > 12 - 10$

$\displaystyle -10y > 2$

$\displaystyle -10y \div (-10) < 2 \div (-10)$

Note that the inequality symbol changes.

$\displaystyle y < -\frac{2}{10}$

$\displaystyle y < -\frac{1}{5}$

or, in interval notation, $\displaystyle \left ( -\infty, -\frac{1}{5} \right )$.

$\displaystyle -\frac{2}{7} y \geq -4$

$\displaystyle -\frac{2}{7} y \cdot\left ( -\frac{7} {2} \right ) \leq -4 \cdot\left ( -\frac{7} {2} \right )$

Note that the inequality symbol changes.

$\displaystyle y\leq \frac{28} {2}$

$\displaystyle y\leq 14$

or, in interval notation, $\displaystyle \left ( -\infty , 14\right ]$.

$\displaystyle 10 > 7-\frac{x}{4} > 5$

$\displaystyle 10 - 7 > 7-\frac{x}{4} - 7 > 5 - 7$

$\displaystyle 3 > -\frac{x}{4} > -2$

$\displaystyle 3 \cdot (-4) < -\frac{x}{4} \cdot (-4) < -2 \cdot (-4)$

Note the switch in the inequality symbol.

$\displaystyle -12< x< 8$.

This can also be written as $\displaystyle (-12, 8)$.

Solve using the properties of inequality, as follows:

$\displaystyle -13 \leq 8-3x \leq 20$

$\displaystyle -13- 8 \leq 8-3x - 8 \leq 20 - 8$

$\displaystyle -21 \leq -3x \leq 12$

$\displaystyle -21 \div (-3) \geq -3x \div (-3) \geq 12 \div (-3)$ 

Note that division by a negative number reverses the symbols.

$\displaystyle 7 \geq x\geq -4$

In interval form, this is $\displaystyle \left [ -4, 7\right ]$.

To solve this you must find the value of $\displaystyle x$.  

The first equation states that $\displaystyle 7x + 6 = 27$. This is a mult-step equation.  The first step is to remove the constant, 6, from the equation; this is done by using the inverse operation, which means you would subtract the 6 from both sides of the equation.  

$\displaystyle 7x + 6 - 6 = 27 -6$

$\displaystyle 7x = 21$

Then divide both sides by the 7 in order to isolate the variable.

$\displaystyle \frac{7x}{7}$ $\displaystyle =\frac{21}{7}$

$\displaystyle x = 3$

Then plug the 3 into the second equation for the value of x.  

$\displaystyle (4 x 3) + 12 = 24$

The first step in the process of solving for $\displaystyle x$ in this problem is to use the distributive property to distribute the $\displaystyle 5$ to what is inside the parentheses.

 $\displaystyle 5x - 25$

The next step is to isolate the variable by using inverse operations. In this example, in order to get rid of the $\displaystyle -25$, you would add $\displaystyle 25$ to both sides of the equation.

$\displaystyle 5x - 25 + 25 = 50 + 25$

$\displaystyle 5x = 75$

The next step is to divide both sides by the coefficient, (the number next to the variable), which in this case is $\displaystyle 5$.    

$\displaystyle \frac{5x}{5} = x$

$\displaystyle \frac{75}{5} = 15$

$\displaystyle x = 15$

In order to solve for the value of $\displaystyle x$ you must isolate the variable.  This is done by subtracting the constant in this equation, which is 12, from both sides of the equation.

$\displaystyle x + 12 - 12 = 30 - 12$

$\displaystyle x = 18$

In order to solve for $\displaystyle x$, we will need the equation to be in terms of $\displaystyle y$, and isolate the variable $\displaystyle x$.

Solve by grouping the $\displaystyle x$ terms together.  Subtract $\displaystyle 7x$ on both sides.

$\displaystyle 2x-(7x) =-2y+7x-(7x)$

$\displaystyle -5x = -2y$

Divide by negative five on both sides.

$\displaystyle \frac{-5x }{-5}= \frac{-2y}{-5}$

The answer is:  $\displaystyle x=\frac{2}{5}y$

Distribute the term on the right side of the equation.

$\displaystyle 2x+18 = -18x+2x$

Combine like terms.

$\displaystyle 2x+18 = -16x$

Subtract $\displaystyle 2x$ on both sides.

$\displaystyle 2x+18 -2x= -16x-2x$

$\displaystyle 18 = -18x$

Divide by negative $\displaystyle 18$ on both sides.

$\displaystyle \frac{18}{-18} = \frac{-18x}{-18}$

The answer is:  $\displaystyle x=-1$

To answer the question, we will solve for y. So, we get

$\displaystyle 12y-3=33$

$\displaystyle 12-3+3=33+3$

$\displaystyle 12y-0=36$

$\displaystyle 12y=36$

$\displaystyle \frac{12y}{12} = \frac{36}{12}$

$\displaystyle y = 3$