SATII数学考试快递——Inequalities
Before you get too comfortable with expressions and equations, we should introduce inequalities. An inequality is like an equation, but instead of relating equal quantities, it specifies exactly how two quantities are not equal. There are four types of inequalities:
x > y: “x is greater than y.”
x < y: “x is less than y.”
x ≥ y: “x is greater than or equal to y.”
x ≤ y: “x is less than or equal to y.”
Solving inequalities is exactly like solving equations except for one very important difference: when both sides of an inequality are multiplied or divided by a negative number, the direction of the inequality switches.
Here are a few examples:
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– 3 < 2y.
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≥ –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,
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First, solve for the upper bound:
Second, solve for the lower bound:
