Inequality Solver Inequality Solver There are many opportunities for mistakes with absolute-value inequalities, so let's cover this topic slowly and look at some helpful pictures along the way. When we're done, I hope you will have a good picture in your head of what is going on, so you won't make some of the more common errors. Once you catch on to how these inequalities work, this stuff really isn't so bad. Let's first return to the original definition of absolute value: "| x | is the distance of x from zero." For instance, since both –2 and 2 are two units from zero, we have | –2 | = | 2 | = 2: With this definition and picture in mind, let's look at some absolute value inequalities. Suppose you're asked to graph the solution to | x | < 3. The solution is going to be all the points that are less than three units away from zero. Look at the number line: The number 1 will work, as will –1; the number 2 will work, as will –2. But 4 will not work, and neither will –4, because they are too far away. Even 3 and –3 won't work (though they're right on the edge). Know More About Stem and Leaf Plot
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But 2.99 will work, as will –2.99. In other words, all the points between –3 and 3, but not actually including –3 or 3, will work in this inequality. Then the solution looks like this: The open circles at the ends of the blue line indicate "up to, but not including, these points." Your book might use parentheses instead of circles. Translating this picture into algebraic symbols, you find that the solution is –3 < x < 3. This pattern for "less than" absolute-value inequalities always holds: Given the inequality | x | < a, the solution is always of the form –a < x < a. Even when the exercises get more complicated, the pattern still holds. Find the absolute-value inequality statement that corresponds to the inequality –2 < x < 4. I first look at the endpoints. Negative two and four are six units apart. Half of six is three. So I want to adjust this inequality so that it relates to –3 and 3, instead of to –2 and 4. To accomplish this, I will adjust the ends by subtracting 1 from all three "sides": –2 < x < 4 –2 – 1 < x – 1 < 4 – 1 –3 < x – 1 < 3 Since the last line above is in the "less than" format, the absolute-value inequality will be of the form "absolute value of something is less than 3". I can convert this nicely to |x–1|<3
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How To Simplify Fractions How To Simplify Fractions A fraction (from Latin: fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, five-eighths, three-quarters. A common or vulgar fraction, such as 1/2, 8/5, 3/4, consists of an integer numerator and a non-zero integer denominatorâ€”the numerator representing a number of equal parts and the denominator indicating how many of those parts make up a whole. An example is 3/4, in which the numerator, 3, tells us that the fraction represents 3 equal parts, and the denominator, 4, tells us that 4 parts equal a whole. Simplifying fractions is often required when your answer is not in the form required by the assignment. As a matter of fact, most math instructor will demand that you always simplify results. Generally speaking, there are two occasions where simplifying your answer may be necessary.
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First, your answer may be a higher equivalent fraction, which is better represented in its simplified or reduced form. Many teachers will insist that you reduce your answer, whenever possible. Also, many operations on fractions will often result in what's called an improper fraction. This is where the numerator is larger than the denominator. To write these answers in their simplest form you will have to convert them to a mixed number. This will show a representation of the Whole Parts and the Fractional Parts. You may remember that an improper fractions is where the numerator has a greater value than that of the denominator. So each time you perform an operation on fractions and your answer ends up as an improper fraction, you will usually need to simplify your answer. The results will be in the form of a mixed number. To convert an improper fraction into a mixed number, just divide the numerator by the denominator. The results will be a whole number part and a fractional part. As you can see, this is a pretty straightforward operation. But keep in mind that if there is no remainder, the answer is the WHOLE NUMBER only. So as you can see, simplifying fractions is no-big-deal. Just follow the steps above, and simplifying fractions will be like a "walk in the park."
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Published on Apr 30, 2012
When we're done, I hope you will have a good picture in your head of what is going on, so you won't make some of the more common errors. Onc...