Inequalities - ACT Math
Card 0 of 423
Solve 
Solve
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 
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
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What values of
make the statement
true?
What values of make the statement
true?
First, solve the inequality
:



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



First, solve the inequality :
Since we are dealing with absolute value, must also be true; therefore:
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Given the inequality
which of the following is correct?
Given the inequality which of the following is correct?
First separate the inequality
into two equations.


Solve the first inequality.



Solve the second inequality.



Thus,
or
.
First separate the inequality into two equations.
Solve the first inequality.
Solve the second inequality.
Thus, or
.
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Simplify
.
Simplify
.
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
:

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 :
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Simplify the following inequality
.
Simplify the following inequality
.
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
:

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 :
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The inequality
is equivalent to which of the following inequalities?
The inequality is equivalent to which of the following inequalities?
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:



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:
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. Solve for 
. Solve for
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!
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!
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Solve the following inequality:

Solve the following inequality:
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 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,
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Solve the following inequality:

Solve the following inequality:
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,

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,
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Solve:

Solve:
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.
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.
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Solve:

Solve:
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 
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
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Each of the following is equivalent to
xy/z * (5(x + y)) EXCEPT:
Each of the following is equivalent to
xy/z * (5(x + y)) EXCEPT:
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.
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.
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Solve for
.

Solve for .


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

When dividing both sides of an inequality by a negative number, you must change the direction of the inequality sign.
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Find is the solution set for x where:

Find is the solution set for x where:
We start by splitting this into two inequalities,
and 
We solve each one, giving us
or
.
We start by splitting this into two inequalities, and
We solve each one, giving us or
.
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Which of the following inequalities defines the solution set for the inequality 14 – 3_x_ ≤ 5?
Which of the following inequalities defines the solution set for the inequality 14 – 3_x_ ≤ 5?
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.
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.
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Solve 3 < 5x + 7
Solve 3 < 5x + 7
Subtract seven from both sides, then divide both sides by 5, giving you –4/5 < x.
Subtract seven from both sides, then divide both sides by 5, giving you –4/5 < x.
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Solve 6_x_ – 13 > 41
Solve 6_x_ – 13 > 41
Add 13 to both sides, giving you 6_x_ > 54, divide both sides by 6, leaving x > 9.
Add 13 to both sides, giving you 6_x_ > 54, divide both sides by 6, leaving x > 9.
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What is the solution set of the inequality
?
What is the solution set of the inequality ?
We simplify this inequality similarly to how we would simplify an equation


Thus 
We simplify this inequality similarly to how we would simplify an equation
Thus
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Solve for the
-intercept:

Solve for the -intercept:
Don't forget to switch the inequality direction if you multiply or divide by a negative.





Now that we have the equation in slope-intercept form, we can see that the y-intercept is 6.
Don't forget to switch the inequality direction if you multiply or divide by a negative.
Now that we have the equation in slope-intercept form, we can see that the y-intercept is 6.
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What is a solution set of the inequality
?
What is a solution set of the inequality ?
In order to find the solution set, we solve
as we would an equation:



Therefore, the solution set is any value of
.
In order to find the solution set, we solve as we would an equation:
Therefore, the solution set is any value of .
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