Solve and graph the solutions of the equation -Ix - 2| + 9 > 6.
To solve the equation, we need to follow the steps:
1) Isolate the absolute value expression by moving other terms to the other side of the inequality sign.
2) Solve the inequality when the expression inside the absolute value is positive.
3) Solve the inequality when the expression inside the absolute value is negative.
4) Combine the solutions and graph them on a number line.
1) First, we isolate the absolute value expression:
-Ix - 2| + 9 > 6
Subtract 9 from both sides:
-Ix - 2| > -3
2) When the expression inside the absolute value is positive:
-Ix - 2 > -3
Add 2 to both sides:
-Ix > -1
Divide both sides by -1, remembering to reverse the inequality sign since we are dividing by a negative number:
Ix < 1
3) When the expression inside the absolute value is negative:
-Ix - 2 < -3
Add 2 to both sides:
-Ix < -1
Divide both sides by -1, remembering to reverse the inequality sign:
Ix > 1
4) Now we can combine the solutions:
Ix < 1 or Ix > 1
To graph the solutions on a number line:
We have two cases to consider:
1) When x is less than 1:
For this case, we use an open circle at 1 and shade everything to the left of 1.
2) When x is greater than 1:
For this case, we use an open circle at 1 and shade everything to the right of 1.
The final graph looks like this:
---|-----o-------------|------------->
-1 1
(shaded) (open circle) (shaded)