Inequalities can be treated like any other equation except when multiplying and dividing by negative numbers. When multiplying or dividing by negative numbers, we just flip the sign of the inequality so that \(\displaystyle >\) becomes \(\displaystyle < \), and vice versa.

\(\displaystyle \frac{-7x-2}{3}>11\)

\(\displaystyle -7x-2>33\)

\(\displaystyle -7x>35\)

\(\displaystyle \frac{-7x}{-7}< \frac{35}{-7}\)

\(\displaystyle x< -5\)

Inequalities can be treated like any other equation except when multiplying and dividing by negative numbers. When multiplying or dividing by negative numbers, we just flip the sign of the inequality so that \(\displaystyle >\) becomes \(\displaystyle < \), and vice versa. When we solve binomials, we must take extra caution because \(\displaystyle (-x)^2=x^2\).

So when we solve inequalities with binomials, we must create two scenarios: one where the value inside of the parentheses is positive and one where it is negative. For the negative scenario, we must flip the sign as we normally do for inequalities.

\(\displaystyle x^2+4x>12\)

\(\displaystyle x^2+4x+4>12+4\)

\(\displaystyle (x+2)^2>16\)

Now we must create our two scenarios:

\(\displaystyle x+2>{\sqrt{16}}\) and \(\displaystyle x+2< -{\sqrt{16}}\)

Notice that in the negative scenario, we flipped the sign of the inequality.

\(\displaystyle x+2>4\) and \(\displaystyle x+2< -4\)

\(\displaystyle x>2\) and \(\displaystyle x< -6\)

Inequalities can be treated like any other equation except when multiplying and dividing by negative numbers. When multiplying or dividing by negative numbers, we just flip the sign of the inequality so that \(\displaystyle >\) becomes \(\displaystyle < \), and vice versa. When we solve binomials, we must take extra caution because \(\displaystyle (-x)^2=x^2\).

So when we solve inequalities with binomials, we must create two scenarios: one where the value inside of the parentheses is positive and one where it is negative. For the negative scenario, we must flip the sign as we normally do for inequalities.

\(\displaystyle x^2-10x< 24\)

\(\displaystyle x^2-10x+25< 24+25\)

\(\displaystyle (x-5)^2< 49\)

Now we must create our two scenarios:

\(\displaystyle x-5< {\sqrt{49}}\) and \(\displaystyle x-5>-{\sqrt{49}}\)

Notice that in the negative scenario, we flipped the sign of the inequality.

\(\displaystyle x-5< 7\) and \(\displaystyle x-5>-7\)

\(\displaystyle x< 12\) and \(\displaystyle x>-2\)

\(\displaystyle 2x+4>64\)

Subtract 4 from both sides:

\(\displaystyle 2x > 60\)

Divide both sides by 2:

\(\displaystyle x > 30\)

\(\displaystyle -7x-4\geq10\)

Add 4 to both sides.

\(\displaystyle -7x\geq14\)

Divide both sides by –7. When dividing by a negative value, we must also change the direction of the inequality sign.

\(\displaystyle x\leq-2\)

\(\displaystyle 9x-27\geq6x\)

Move like terms to the same sides:

\(\displaystyle 9x-6x\geq27\)

Combine like terms:

\(\displaystyle 3x\geq27\)

Divide both sides by 3:

\(\displaystyle x\geq9\)

Inequalities can be treated like any other equation except when multiplying and dividing by negative numbers. When multiplying or dividing by negative numbers, we just flip the sign of the inequality so that \(\displaystyle >\) becomes \(\displaystyle < \), and vice versa.

\(\displaystyle 5-3x\leq14\)

\(\displaystyle -3x\leq9\)

\(\displaystyle \frac{-3x}{-3}\geq\frac{9}{-3}\)

\(\displaystyle x\geq-3\)