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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: 1. -(x+4)>=10 == -x-4>=10 == x<-14 2. x+4>=10 == x>=6 sooo: x<=-14 OR x>=6.
Thank you very much! My student says Yippee!
What about |9x-5| greater than or equal to 4
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).