+0

# help im confused

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What values of x satisfy |x - 4| + |x + 4| <= 10\$?

ive found out the answers of      [4,5] or [-5,-4]

and dont understand where i went wrong in the problem.

Can someone explain?

Aug 28, 2018

#1
+100516
+2

|x - 4| + |x + 4| <= 10

We have these  equations

x - 4  + x + 4   <=  10

2x  <=  10

x  <= 5

And

x - 4  + x + 4   >=  -10

2x   >= -10

x >= -5

So.....the solution  is   [-5 , 5  ]

Here's the graph of the solution  :  https://www.desmos.com/calculator/7ar4jqmtg3

Aug 28, 2018
#2
+22172
+3

What values of x satisfy |x - 4| + |x + 4| <= 10 ?

$$\huge |x + 4| + |x - 4| \le 10$$

Here we have two different amounts.

To dissolve them, we must make a double case distinction.

Usually we did this one after the other.
For reasons of space, we start with the first case in which the content of the left amount
is greater than or equal to zero.

$$\begin{array}{|rcll|} \hline & \underline{x\ge -4:}& \\\\ & x+4 + |x-4 | \le 10 \\ \\ \underline{\text{for } x\ge 4:} && \underline{\text{for } x \lt 4: } \\\\ x+4+x-4 \le 10 && x+4-(x-4) \le 10 \\ 2x \le 10 && 8 \le 10 \\ \boxed{ x \le 5 } && \\ \hline \end{array}$$

The other case was $$x \lt -4$$.
In this area the content of the right amount is
always negative. A further case distinction is therefore not necessary here.

$$\begin{array}{|rcll|} \hline & \underline{x \lt -4:} \\\\ & -(x+4) - (x-4) \le 10 \\ \\ & -x-4-x+4 \le 10 \\ & -2x \le 10 \quad & \quad :(-2)\\ & x \ge \dfrac{10}{-2} \\ & \boxed{x \ge -5} \\ \hline \end{array}$$

Values of x in interval notation: [-5 , 5  ]

Aug 29, 2018