Inequalities - ACT Math

Solve the inequality by adding 3 to both sides to get x < 12. Since it is absolute value, x – 3 > –9 must also be solved by adding 3 to both sides so: x > –6 so combined.

Absolute values have two answers: a positive one and a negative one. Therefore,

-1 ≤ x – 5≤ 1 and solve by adding 5 to all sides to get 4 ≤ x ≤ 6.

Absolute value problems always have two sides: one positive and one negative.

First, take the problem as is and drop the absolute value signs for the positive side: z – 3 ≥ 5. When the original inequality is multiplied by –1 we get z – 3 ≤ –5.

Solve each inequality separately to get z ≤ –2 or z ≥ 8 (the inequality sign flips when multiplying or dividing by a negative number).

We can verify the solution by substituting in 0 for z to see if we get a true or false statement. Since –3 ≥ 5 is always false we know we want the two outside inequalities, rather than their intersection.

3_w_/2 will become greater than 1 as soon as w is greater than two thirds. It will likewise become less than –1 as soon as w is less than negative two thirds. All the other options always return values between –1 and 1.

To solve this problem, add the two equations together:

The only answer choice that satisfies this equation is 0, because 0 is less than 4.

Absolute value is the distance from the origin and is always positive.

So we need to solve and which becomes a bounded solution.

Adding 3 to both sides of the inequality gives and or in simplified form

First, solve the inequality :

Since we are dealing with absolute value, must also be true; therefore:

First separate the inequality into two equations.

Solve the first inequality.

Solve the second inequality.

Thus, or .

Simplifying an inequality like this is very simple. You merely need to treat it like an equation—just don't forget to keep the inequality sign.

First, subtract from both sides:

Then, divide by :

For a combined inequality like this, you just need to be careful to perform your operations on all the parts of the inequality. Thus, begin by subtracting from each member:

Next, divide all of the members by :

In order to simplify an inequality, we must bring the unknown () values on one side and the integers on the other side of the inequality:

We must put all of the like terms together on either side of the inequality symbol. First, we need to subtract the to the right side and add to the left side to get all of the terms with to the right side of the inequality and all of the integers to the left side.

We solve for by dividing by .

That leaves us with , which is the same as . Remember, you only flip the direction of the inequality if you divide by a negative number!

To solve an equality that has addition, simply treat it as an equation. Remember, the only time you have to do something to the inquality is when you are multiplying or dividing by a negative number.

Subrtract 4 from each side. Thus,

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,

First, we want to group all of our like terms. I will move all of my integers to the left side of the inequality.

Since we are not dividing by a negative sign, we do not have to flip the inequality.

The first thing that we have to do is deal with the absolute value. We simply remove the absolute value by equating the left side with the positive and negative solution (of the right side). When we include the negative solution, we must flip the direction of the inequality. Shown explicitly:

Now, we simply solve the inequality by moving all of the integers to the right side, and we are left with: This reduces down to

Choice a is equivalent because we can say that technically we are multiplying two fractions together: (xy)/z and (5(x + y))/1. We multiply the numerators together and the denominators together and end up with xy (5x + 5y)/z. xy (5y + 5x)/z is also equivalent because it is only simplifying what is inside the parentheses and switching the order- the commutative property tells us this is still the same expression. 5x²y + 5xy²/z is equivalent as it is just a simplified version when the numerators are multiplied out. Choice 5x² + y²/z is not equivalent because it does not account for all the variables that were in the given expression and it does not use FOIL correctly.

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

We start by splitting this into two inequalities, and

We solve each one, giving us or .

To solve this inequality, you should first subtract 14 from both sides.

This leaves you with –3_x_ ≤ –9.

In the next step, you divide both sides by –3, remembering to flip the inequality sign when you do this.

This leaves you with the solution x ≥ 3.

If you selected x ≤ 3, you probably forgot to flip the sign. If you selected one of the other solutions, you may have subtracted incorrectly.