Solving an Absolute Value Inequality with Two Terms

In summary: So if we want to find the smallest value of x that solves the inequality, we can just take the square of the distance between x and 5 and it will be 1.
  • #1
projection
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0

Homework Statement



[tex]\left|x-5\right|\leq\left|x+3\right|[/tex]

i know how to slove abs inequalities when there is one. like [tex]\left|x-5\right|\leq5[/tex]. put x-5 between 5 and -5. then solve for x.

i don't know how to start this.

can someone provide some insight.
 
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  • #2
projection said:

Homework Statement



[tex]\left|x-5\right|\leq\left|x+3\right|[/tex]

i know how to slove abs inequalities when there is one. like [tex]\left|x-5\right|\leq5[/tex]. put x-5 between 5 and -5. then solve for x.

i don't know how to start this.

can someone provide some insight.
pretty rusty myself with abs inequalities, but can you just do the same as what you mentioned [tex]\left|x-5\right|\leq5[/tex].
i.e:
[tex]\left|x-5\right|\leq\left|x+3\right|[/tex]
=

[tex]x-5 \leq\ x+3 \leq\ x+5[/tex]
and then solve from there?

as I said, I'm rusty on these things so apologies if I'm leading you up the garden path!
 
Last edited:
  • #3
A method I prefer is to solve the associated equation first!
If |x- 5|= |x+ 3| then either x- 5= x+ 3 or x- 5= -(x+ 3). The first reduces to the equation -5= 3 which is NEVER true and so does not give a solution to the equation. The second is the same as x- 5= -x- 3 or, adding x and 5 to both sides, 2x= 2 so x= 1

Now, the point is that x= 1, where |x-5|= |x+ 3| separates |x-5|< |x+3| and |x- 5|> |x+ 3|. check one value of x on each side of 1 (x= 0 and x= 2 are simple) to see on which side |x-5|< |x+3|.

You might also remember that |x- a| can be interpreted as "distance from x to a". |x- 5| can be interpreted as "distance from x to 5" and |x+3|= |x-(-3)| as "distance from x to -3". Asking "where is |x-5|< |x+3|?" is the same as asking "what points on the line are closer to 5 than to -3?". You might notice that, geometrically, x= 1 is exactly half way between 5 and -3.
 
  • #4
Just so you get exposed to this, another way to think about the problem is follows:

[tex]|x - 5| \leq |x + 3| \implies (x-5)^2 \leq (x+3)^2 \implies x^2-10x+25 \leq x^2+6x+9 \implies 16 \leq 16x \implies 1 \leq x[/tex]

Notice that the x2 cancels out so we only have one solution.
 
  • #5
A little off-topic (and this has prob been well-documented already, but apologies: I am new here!), but I wrote:
[tex]x-5\leq\x+3\leq\x-5[/tex]
then realized the error and edited it to:
[tex]x-5\leq\x+3\leq\x+5[/tex]
but it keeps showing up as the original on screen. If you click on it for the LaTeX code reference, it shows it as the corrected version.


why is this?

edit: now the x is missing! what am I doing wrong here?
 
  • #6
That is because in your most recent LaTeX, you have a backslash before the x, which shouldn't be there. For your other problem, make sure you erase the browser cache when you refresh. Most times, on Windows, it is Shift+F5 or Ctrl+F5 instead of just F5.
 
  • #7
cheers. wanted to clear that up before I make any more screw-ups.
 
  • #8
Tedjn said:
Just so you get exposed to this, another way to think about the problem is follows:

[tex]|x - 5| \leq |x + 3| \implies (x-5)^2 \leq (x+3)^2 \implies x^2-10x+25 \leq x^2+6x+9 \implies 16 \leq 16x \implies 1 \leq x[/tex]

Notice that the x2 cancels out so we only have one solution.

is there an explanation as to why this would work?
 
  • #9
It works really because each statement follows from the previous one. If you want a more conceptual way to think about it, I think HallofIvy's idea about picturing absolute value as distance on a number line is the best choice. In any case, from that picture, basically this takes the square of that distance, and finds what values of x work.
 
  • #10
It works because |x|2= x2 for any real number, x.
 

Related to Solving an Absolute Value Inequality with Two Terms

1. How do I solve an absolute value inequality with two terms?

To solve an absolute value inequality with two terms, first isolate the absolute value expression on one side of the inequality. Then, set up two separate equations, one with the positive value of the expression and one with the negative value. Solve each equation separately and combine the solutions to find the overall solution.

2. What is the difference between solving an absolute value inequality and solving an absolute value equation?

The main difference between solving an absolute value inequality and solving an absolute value equation is that an inequality has multiple possible solutions, whereas an equation has only one solution.

3. Can I use the same steps to solve any absolute value inequality with two terms?

Yes, the same steps can be used to solve any absolute value inequality with two terms. However, the specific numbers and expressions in the inequality may vary, resulting in different solutions.

4. How do I know which solution is correct when solving an absolute value inequality with two terms?

When solving an absolute value inequality with two terms, it is important to check your solutions by plugging them back into the original inequality. The solution(s) that make the inequality true are the correct solution(s).

5. Are there any common mistakes to avoid when solving an absolute value inequality with two terms?

One common mistake to avoid when solving an absolute value inequality with two terms is forgetting to change the direction of the inequality symbol when dividing or multiplying by a negative number. It is also important to carefully distribute any negative signs when expanding the inequality.

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