You can put this solution on YOUR website! Your answers are correct. Good job!
.
In the way of discussion you can solve problems like this by using the plus and minus
quantities inside the absolute value signs. Work them like equations, except for one
exception ... if you multiply or divide both sides by a negative quantity, you reverse
the direction of the inequality sign.
.
Let's apply these techniques to this problem.
.
First, the quantity inside the absolute value signs is (x - 1.5).
.
Next put a plus sign in front of this quantity and you get the inequality:
.
+(x - 1.5) <= 3
.
Apply the procedures just as you would a equation. Begin by adding 1.5 to both sides
to remove the 1.5 from the left side. This makes the equation become:
.
x <= 3 + 1.5
.
add the terms on the right side and you get:
.
x <= 4.5 ... the same answer that you got.
.
Now for the second case. Use the quantity inside the absolute value signs, only precede
it by a negative sign. This makes the inequality:
.
-(x - 1.5) <= 3
.
The negative sign in front of the parentheses means that you are multiplying the quantity
in parentheses by -1. This results in:
.
-x + 1.5 <= 3
.
Get rid of the 1.5 on the left side by subtracting 1.5 from both sides. This simplifies
the equation to:
.
-x <= 3 - 1.5
.
On the right side of the inequality the two terms combine to 1.5 This makes the inequality
become:
.
-x <= 1.5
.
But we need to solve for +x. So multiply both sides by -1. (Don't forget the rule that
if you multiply or divide both sides by a negative number, you need to reverse the
direction of the inequality sign.) When you do these to things (multiply by -1 and
reverse the direction) you get:
.
x >= -1.5
.
This also is the same as your answer.
.
You can write the two answers in a "one-liner". When you say that x must be greater
than or equal to -1.5 that means it must be at or to the right of 1.5 on the number
line. But you also know that x must be equal to or to the left of +4.5 on the number line.
This can be written as:
.
- 1.5 <= x <= +4.5
.
Just to help you convince yourself that this is correct, select a number between -1.5 and
+4.5 set x equal to that number. For example, 0 is between those two limits. Setting
x equal to zero makes the original inequality:
.
.
This reduces to:
.
.
But the absolute value of -1.5 is +1.5 and so the inequality is:
.
.
This is true, so that gives some confidence that we have the right answer. You can do
the same sort of analysis by picking a value for x that is less than -1.5 (say for example,
that x = -5). Substitute that value for x in the original problem and ensure yourself
that it does NOT satisfy the inequality. Do the same sort of analysis by letting x
be a value greater than +4.5 such as x = +5. Substitute +5 for x in the original
inequality and you should see that you get a value on the left side that is greater
than 3 ... meaning that the inequality does not work for values of x greater than 4.5.
.
Hope this discussion gives you a logical way to work problems of this type and how to
check to build confidence that your answer is correct.
.
Don't let the above discussion confuse you. You seem to know how to work problems
of this type. Keep up the good work.
(