In order to isolate the variable, we need to remove the coefficient. Since the operation between \(\displaystyle \tiny \small \frac{5}{7}\) and \(\displaystyle \small x\) is multiplication, we may want to divide both sides of the inequality by \(\displaystyle \tiny \small \frac{5}{7}\). Although this is a valid step, in order to simplify matters, instead of dividing by \(\displaystyle \tiny \small \frac{5}{7}\), we can multiplying by \(\displaystyle \small \frac{7}{5}\).

(Note: dividing by \(\displaystyle \frac{5}{7}\) is exactly the same as multplying by the reciprocal, \(\displaystyle \frac{7}{5}\).) 

Thus,

\(\displaystyle \small x> \frac{15}{1} \cdot\frac{7}{5}\) and after multiplying and simplifying, we obtain \(\displaystyle \small x>21\).  

Simplify \(\displaystyle 3x-2-x>-2x+6\) by combining like terms to get \(\displaystyle 2x-2>-2x+6\).

Then, add \(\displaystyle 2\) and \(\displaystyle 2x\) to both sides to separate the \(\displaystyle x\)'s and intergers. This gives you \(\displaystyle 4x>8\).

Divide both sides to get \(\displaystyle x>2\). Since we didn't divide by a negative number, there is no need to reverse the sign.

\(\displaystyle \frac{7}{9} x > \frac{5}{3}\)

\(\displaystyle \frac{7}{9} x \div \frac{7}{9}> \frac{5}{3} \div \frac{7}{9}\)

\(\displaystyle \frac{7}{9} x \cdot \frac{9}{7}> \frac{5}{3}\cdot \frac{9}{7}\)

Cross-cancel:

\(\displaystyle x> \frac{5}{1}\cdot \frac{3}{7}\)

\(\displaystyle x> \frac{15}{7}\)

or, in interval form, \(\displaystyle \left ( \frac{15}{7},\infty \right )\).

Eliminate fractions by multiplying by the least common denominator - \(\displaystyle LCM(3,6,9) = 18\).

\(\displaystyle 18 \cdot \left (\frac{2}{9} x + \frac{1}{3} \right ) < 18 \cdot \frac{5}{6}\)

\(\displaystyle \frac{18}{1} \cdot \frac{2}{9} x + \frac{18}{1} \cdot \frac{1}{3} \right ) < \frac{18}{1} \cdot \frac{5}{6}\)

Cross-cancel:

\(\displaystyle \frac{2}{1} \cdot \frac{2}{1} x + \frac{6}{1} \cdot \frac{1}{1} \right ) < \frac{3}{1} \cdot \frac{5}{1}\)

\(\displaystyle 4x + 6 < 15\)

\(\displaystyle 4x + 6 - 6 < 15 - 6\)

\(\displaystyle 4x < 9\)

\(\displaystyle 4x \div 4 < 9 \div 4\)

\(\displaystyle x < \frac{9}{4}\)

\(\displaystyle -\frac{7}{10} < -\frac{2}{5} x \leq \frac{1}{10}\)

\(\displaystyle - \frac{5}{2} \cdot \left (-\frac{7}{10} \right )> - \frac{5}{2} \cdot \left ( -\frac{2}{5}x \right ) \geq - \frac{5}{2} \cdot \frac{1}{10}\)

Note the switch in inequality symbols when the numbers are multiplied by a negative number.

\(\displaystyle \frac{5}{2} \cdot \frac{7}{10} > x \geq - \frac{5}{2} \cdot \frac{1}{10}\)

Cross-cancel:

\(\displaystyle \frac{1}{2} \cdot \frac{7}{2} > x \geq - \frac{1}{2} \cdot \frac{1}{2}\)

\(\displaystyle \frac{7}{4} > x \geq - \frac{1}{4}\)

or, in interval form, \(\displaystyle \left [ - \frac{1}{4},\frac{7}{4} \right )\)

Subtract 4 from both sides.  Then subtract 9x:

\(\displaystyle -6x>-12\)

Next divide both sides by -6.  Don't forget to switch the inequality because of the negative sign!

\(\displaystyle x< 2\)

To solve the inequality, subtract \(\displaystyle x\) and add 12 to both sides to separate the \(\displaystyle x\) from the integers:

\(\displaystyle 2x>14\)

Divide both sides by 2:

\(\displaystyle x>7\)

Note: The inequality sign is only flipped when dividing by negative numbers.

First, combine the like terms on the righthand side of the inequality to get \(\displaystyle 3x-5>6x+1\).

Then, subtract \(\displaystyle 3x\) and \(\displaystyle 1\) from both sides to get \(\displaystyle -6>3x\).

Finally, divide both sides by \(\displaystyle 3\): \(\displaystyle x< -2\)

Isolate all the terms with \(\displaystyle x\) on one side and the other terms on the other side and solve for \(\displaystyle x\). 

\(\displaystyle -3x+2 < x+4\)

First subtract the an x from both sides of the inequality. The subtract 2 from each side. This results in the following inequality.

\(\displaystyle -4x< 2\)

Here, we need to divide both sides by \(\displaystyle -4\). However, whenever we divide or multiply and inequality by a negative number, we have to also change the direction of the inequality.

The final answer becomes

\(\displaystyle x>-\frac{1}{2}\).

To solve an inequality, isolate the variable on one side with all other constants on the other side. To accomplish this, perform opposite operations to manipulate the inequality.

First, isolate the x by multiplying each side by two.

Whatever you do to one side you must also do to the other side.

\(\displaystyle \frac{2x}{{\color{Blue} 2}}\leq\frac{6}{{\color{Blue} 2}}\)

This gives you:

\(\displaystyle x\leq3\)

The answer, therefore, is \(\displaystyle (-\infty ,3]\).