ACT Math - How to find the solution to an inequality with division

Find is the solution set for x where:

$\left | 5x-3\right |>12$

$-\frac{9}{5}< x< 3$

$-3< x< \frac{9}{5}$

or $x< \frac{9}{5}$

$x< -\frac{9}{5}$

or $x< -\frac{9}{5}$

Explanation

We start by splitting this into two inequalities, and

We solve each one, giving us or $x<\frac{-9}{5}\right$ .

Simplify the following inequality

$4 - 3x \ge 22 + 2x$ .

$x \le \frac{-18}{5}$

$x \ge \frac{-18}{5}$

$x \ge \frac{18}{5}$

$x \le \frac{2}{7}$

$x \ge \frac{14}{5}$

Explanation

For the most part, you can treat inequalities just like equations. (It is not exact, as you will see below.) Thus, start by isolating your variables. Subtract from both sides:

$4 - 5x \ge 22$

Next, subtract from both sides:

$- 5x \ge 18$

Finally—here you need to be careful—divide by . When you divide or multiply by a negative value in inequalities, you need to flip the inequality sign.

Thus, you get:

$x \le \frac{-18}{5}$

Solve the following inequality:

Explanation

To solve, simply treat it as an equation.

This means you want to isolate the variable on one side and move all other constants to the other side through opposite operation manipulation.

Remember, you only flip the inequality sign if you multiply or divide by a negative number.

Thus,

$3x>27\Rightarrow \frac{3x}{3}>\frac{27}{3}\Rightarrow x>9$

Solve the following inequality:

$x>\frac{1}{2}$

$x<\frac{1}{2}$

Explanation

To solve, simply solve as though it is an equation.

The goal is to isolate the variable on one side with all other constants on the other side. Perform the opposite operation to manipulate the inequality.

Only when dividing or multiplying by a negative number, will you have to flip the inequality sign.

$x=\frac{2}{4}$

$x=\frac{1}{2}$

$x>\frac{1}{2}$

Solve for .

$\small 14-2x\geq 22$

$\small x\leq-4$

$\small x\leq4$

$\small x\geq4$

$\small x\geq-4$

Explanation

$\small 14-2x\geq22$

$\small -2x\geq8$

When dividing both sides of an inequality by a negative number, you must change the direction of the inequality sign.

$\small x\leq-4$

What is the solution to the given inequality:
$-3x\geq 12$

$x\leq -4$

$x\geq -4$

$x\leq4$

Explanation

When solving an inequality in which you have to mulitiply or divide by a negative number, you must "flip" the direction of the inequality. Other than that, solve it nomrally.

Thus the first and only step we have is to divide by and since that number is negative, we "flip" the inequality.