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well a very simple explanation:
the |x+4| means the absolute value of x+4; therefore if it's negative it mutliplies it by -1,
and if it's positive it leaves it as it is.
for the answer, you need to consider BOTH cases:
sooo: x<=-14 OR x>=6.
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do the same thing.
Consider |x - 5| = 2.
On the number line, this literally means all numbers x
that are 2 units away from the number 5.
Draw the number line and circle 5.
What are the numbers 2 units away from it?
They are 3 and 7.
You c an help your student by drawing a soccer player at 5.
Then the soccer player kicks the ball 2 units left or two units right.
For |x – 5| > 2,
this means that the soccer player kicks the ball so hard that it flies past 3 and 7
The ball goes beyond the number 7 to the right.
The ball can also go beyond the number 3 to the left.
So the solution is x > 7 or x < 3.
For |x – 5| < 2,
the soccer player kicks the ball softly enough so that it doesn’t go past 3 or 7.
In fact, it stays bounded between the goalies at 3 and 7.
So 3 < x < 7 is the solution.
For your problem, we have |x + 4| > 10.
If your student knows negative numbers, then |x + 4| = |x – (-4)| > 10.
So the soccer player now stands on the number -4 and kicks it more than 10 yards.
The goalies are standing 10 yards away from the player, they’re standing at
6 and -14.
So the soccer player kicks it past the guy at 6 or past the guy at -14.
So x > 6 or x < -14.
Thank you very much. I will do this for the student.
For |9x – 5| < 4, that’s trickier.
but notice now that we understand how to solve inequalities of the form
|x – A| < B. The player stands at A, kicks the ball softer than B units.
The goalies are located at A + B and A – B.
If we can boil |9x – 5| < 4 to something equivalent of the form |x – A| < B,
then we can use our previous understanding.
To go from |9x – 5| < 4 to |x – A| < B,
we can just divide both sides by 9.
This gives us an equivalent inequality |x – (5/9)| < (4/9).
Then the soccer player stands at 5/9 and kicks it softly so that it’s between the goalies,
located (5/9) + (4/9) and (5/9) – (4/9).