You can put this solution on YOUR website! You can treat this somewhat like you do when you solve an equation ... whatever you do to
one section of the inequality, you must also do to the other two sections. And you also
need to follow the rule that if you divide or multiply all the sections by a NEGATIVE
quantity, you must reverse the directions of the inequality signs. And you should always
check your answer by using some convenient values for each section. Let's apply these rules
to your problem. You were given:
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We want to solve for x. So let's start by eliminating the +1 in the middle section.
We do that by subtracting 1 from the middle section, but when we do that we must also subtract
1 from the other two sections of the inequality. The subtraction is:
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and this simplifies to:
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Since we are solving for x, we need to divide the center section by 2. But then we must
divide all three sections by 2. So the inequality becomes:
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and this simplifies to:
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and if it helps you to visualize it on the number line better, you can convert the fractions
to decimals and the equation becomes:
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This says that on the number line x must be greater than -1.5 and less than or equal to
+2.5
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Since zero is a number within these two limits, let's return to the original problem and
let x be zero. Then we can see if the inequality still holds true.
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Substitute 0 for x and the inequality becomes:
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Is this true? Sure ... 1 is between -2 and +6 on the number line. With this spot check
we can tell that the inequality holds for at least one value of x between -1.5 and +2.5.
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Let's now select a value of x outside the limits of our answer. For instance, let's chose
x = -2. This should NOT work in the original inequality. Plug it in for x in the original
inequality.
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Notice that this does NOT work because -3 is not greater than -2.
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and then we need to do a similar check for x greater than +2.5. Suppose we let x be +3.
The original inequality would become:
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and the middle section becomes 6 + 1 so that the inequality is:
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This is not true because 7 is not less than or equal to 6.
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Our solution looks pretty good. We did not have to multiply or divide all three sections
of the inequality by a negative number so we did not have to reverse the direction of
the inequality arrows. But don't forget that rule. In some problems you will have to make
use of it.
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Hope this helps you to understand inequalities a little better.
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