nevergonnahelp
Answered

Discover a wealth of knowledge at Westonci.ca, where experts provide answers to your most pressing questions. Discover in-depth answers to your questions from a wide network of experts on our user-friendly Q&A platform. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.

solve the inequality |x+6-|3x+6||+|x+2|-x-4 less then or equal to 0

Sagot :

LammettHash

It looks like the inequality is

[tex]\bigg|x + 6 - |3x + 6|\bigg| + |x + 2| - x - 4 \le 0[/tex]

or equivalently,

[tex]\bigg|x + 6 - 3 |x + 2|\bigg| + |x + 2| - x - 4 \le 0[/tex]

Recall the definition of absolute value:

[tex]|x| = \begin{cases} x & \text{if } x \ge 0 \\ -x & \text{if } x < 0 \end{cases}[/tex]

By this definition, we have

[tex]|x + 2| = \begin{cases} x + 2 & \text{if } x + 2 \ge 0 \\ -(x + 2) & \text{if } x + 2 < 0 \end{cases} = \begin{cases} x + 2 & \text{if } x \ge -2 \\ -x - 2 & \text{if } x < -2 \end{cases}[/tex]

Suppose [tex]x < -2[/tex]. Then

[tex]\bigg| x + 6 - |3x + 6| \bigg| + |x + 2| - x - 4 \\\\ = |x + 6 - 3 (-x - 2)| + (-x - 2) - x - 4 \\\\ = |4x + 12| - 2x - 6[/tex]

By definition of absolute value,

[tex]|4x + 12| = 4 |x + 3| = \begin{cases} 4x + 12 & \text{if } x \ge -3 \\ -4x - 12 & \text{if } x < -3 \end{cases}[/tex]

so we further suppose that [tex]x < -3[/tex]. Then

[tex]\bigg| x + 6 - |3x + 6| \bigg| + |x + 2| - x - 4 \\\\ = |4x + 12| - 2x - 6 \\\\ = (-4x - 12) - 2x - 6 \\\\ = -6x - 18 \le 0 \\\\ \implies x \ge -3[/tex]

but this is a contradiction, so this case gives no solution.

If instead we have [tex]-3 \le x < -2[/tex], then

[tex]\bigg| x + 6 - |3x + 6| \bigg| + |x + 2| - x - 4 \\\\ = |4x + 12| - 2x - 6 \\\\ = (4x + 12) - 2x - 6 \\\\ = 2x + 6 \le 0 \\\\ \implies x \le -3[/tex]

and [tex]-3 \le x[/tex] and [tex]x \le -3[/tex] means that [tex]\boxed{x = -3}[/tex].

Now suppose [tex]x \ge -2[/tex]. Then

[tex]\bigg| x + 6 - |3x + 6| \bigg| + |x + 2| - x - 4 \\\\ = |x + 6 - 3 (x + 2)| + (x + 2) - x - 4 \\\\ = |-2x| - 2 \\\\ = 2 |x| - 2[/tex]

If we further suppose that [tex]-2 \le x < 0[/tex], then

[tex]\bigg| x + 6 - |3x + 6| \bigg| + |x + 2| - x - 4 \\\\ = 2 |x| - 2 \\\\ = -2x - 2 \le 0 \\\\ \implies x \ge -1[/tex]

Otherwise, if [tex]x \ge 0[/tex], then

[tex]\bigg| x + 6 - |3x + 6| \bigg| + |x + 2| - x - 4 \\\\ = 2 |x| - 2 \\\\ = 2x - 2 \le 0 \\\\ \implies x \le 1[/tex]

Taken together, this case gives the solution [tex]\boxed{-1 \le x \le 1}[/tex]

So, the overall solution to the inequality is the set

[tex]\boxed{\left\{ x \in \mathbb R : x = -3 \text{ or } -1 \le x \le -1 \right\}}[/tex]

Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Westonci.ca is committed to providing accurate answers. Come back soon for more trustworthy information.