All ACT Math Resources
Example Question #6 : Inequalities
Solve |x – 5| ≤ 1
None of the answers are correct
-1 ≤ x ≤ 1
4 ≤ x ≤ 6
0 ≤ x ≤ 1
x ≤ 4 or x ≥ 6
4 ≤ x ≤ 6
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.
Example Question #1 : How To Find The Solution To An Inequality With Addition
All real numbers
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
Example Question #7 : Inequalities
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 .
Example Question #5 : Inequalities
What values of x make the following statement true?
|x – 3| < 9
–3 < x < 9
6 < x < 12
–12 < x < 6
x < 12
–6 < x < 12
–6 < x < 12
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.
Example Question #27 : Inequalities
If –1 < w < 1, all of the following must also be greater than –1 and less than 1 EXCEPT for which choice?
3w/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.
Example Question #2 : How To Find The Solution To An Inequality With Addition
Solve for .
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.
Example Question #3 : How To Find The Solution To An Inequality With Addition
If and , then what is the value of ?
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.
Example Question #232 : Gre Quantitative Reasoning
What values of make the statement true?
First, solve the inequality :
Since we are dealing with absolute value, must also be true; therefore:
Example Question #4 : How To Find The Solution To An Inequality With Addition
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 :
Example Question #5 : How To Find The Solution To An Inequality With Addition
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 :