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### Course: Algebra (all content)>Unit 8

Lesson 3: Solving absolute value inequalities

# Solving absolute value inequalities: no solution

Sal solves the inequality |y|+22 ≤ 13.5 to find that it has no solution. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

• Can you put a negative sign outside the absolute value sign?
• Good question! Yes you can. The graph f(x) = -lxl looks exactly like f(x) = lxl except it is flipped on the x-axis. In other words everything will be negative (or zero) no matter what number is inside.
I hope this helps!
• When Sal says there is no solution, he means, there is no solution in real numbers? its possible exists a solution in complex numbers to solve the inequality?
• No. Even when we extend the notion of absolute values to the complex numbers, the values never become negative. Just like with the imaginary numbers themselves, you'd have to artificially define a new number with an absolute value of -1 if you thought that finding formal solutions of equations like this were worth having.
• What if the inequality has a negative absolute value, but the variable is greater than? Such as k>-5 Is this still no solution? Because there are positive numbers that are above -5, they're just not -5.
• Yes, `| k | > -5` has a solution set, but `| k | < -5` does not since the absolute value cannot be negative.
• how can I solve double absolute value equations?
• Hi,
I would just like to know if there are any practice questions associated with this video. If so, please give me a link. Thanks!
• I can not figure out how to solve an inequality with two numbers outside the absolute value sign. For instance, how would I solve 4|2w+3| - 7< 9?
• Can you make a video explaining how to solve inequalities with the two absolute values on the same side?
Such as: | x+1 | + | x+1 | less than or equal to 2
• I am also having trouble solving an inequality with two absolute value functions. I am not aware of any videos about this topic on Khan Academy therefore I am enquiring as to whether anyone has any websites that explain the methods for solving such inequalities. the website rynkwn suggested has an internal error. the questions i am trying to solve is: find the set of values of x for which | x – 1 | > | 2x – 1 |. one suggested method is squaring both sides...
(1 vote)
• As you may have seen from other replies, for solving such problems you have to divide the equation into "regimes", based on the expression(s) of x that are enclosed in absolute value brackets.

Based on your equation, we have three regimes:
(i) x >= 1 (ii) 1/2 <= x < 1 (iii) x < 1/2

For (i): the equation becomes x - 1 > 2x - 1, giving x < 0
However the assumption was x >= 1. Since the assumption is inconsistent with the solution, x>=1 is not a solution.

(ii) For this regime, the equation becomes: 1-x > 2x - 1
This gives the solution x < 2/3; If we combine with our regime assumption, we get the solution set as 1/2 <= x < 2/3

(iii) For this regime, our equation is: 1 - x > 1 - 2x
This gives the solution set x > 0. If we combine this with our regime assumption, we get the solution set as: 0 < x < 1/2

THerefore the final solution set, combining results of (ii) and (iii) is:
0 < x < 2/3. You can take random values within this regime to make sure the solution set satisfies the inequality.