Solve for x in the inequality ≥ –2.
Notice that in the last example, the inequality had to be reversed. Another way to express the solution is x ≥ –2. To help remember that multiplication or division by a negative number reverses the direction of the inequality, remember that if x > y, then –x < –y, just as 5 > 4 and –5 < –4. Intuitively, this idea makes sense, and it might help you remember this special rule of inequalities. Absolute Value and Inequalities When absolute values are included in inequalities, the solutions come in two varieties.If the absolute value is less than a given quantity, then the solution is a single range, with a lower and an upper bound. For example,
Solve for x in the inequality |2x – 4| ≤ 6.
First, solve for the upper bound:Second, solve for the lower bound:Now, combine the two bounds into a range of values for x. –1 ≤ x ≤ 5 is the solution.The other solution for an absolute value inequality involves two disjoint ranges: one whose lower bound is negative infinity and whose upper bound is a real number, and one whose lower bound is a real number and whose upper bound is infinity. This occurs when the absolute value is 美国GREater than a given quantity. For example,
Solve for x in the inequality |3x + 4| > 16.
First, solve for the upper range:Then, solve for the lower range:Now combine the two ranges to form the solution, which is two disjoint ranges: –∞ < x < –20 3 or 4 < x < ∞. When working with absolute values, it is important to first isolate the expression within absolute value brackets. Then, and only then, should you solve separately for the cases in which the quantity is positive and negative. Ranges Inequalities are also used to express the range of values that a variable can take. A < x < b means that the value of x is 美国GREater than a and less than b. Consider the following word-problem example:
A very complicated board game has the following recommendation on the box: “This game is only appropriate for people older than 40 but no older than 65.” What is the range of the age of people for which the board game is appropriate?
Let a be the age of people for which the board game is appropriate. The lower bound of a is 40, and the upper bound is 65. The range of a does not include its lower bound (it is appropriate for people “older than 40”), but it does include its upper bound (“no older than 65”, i.e., 65 is appropriate, but 66 is not). Therefore, the range of the age of people for which the board game is appropriate can be expressed by the inequality: Here is another example:
A company manufactures car parts. As is the case with any system of mass production, small errors occur on virtually every part. The key for this company to succeed in making viable car parts is to keep the errors within a specific range. The company knows that a particular piece they manufacture will not work if it weighs less than 98% of its target weight or more than 102% of its target weight. If the target weight of this piece is 21.5 grams, in what range of weights must the piece measure for it to function?英语作文
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