Distribute the expression. 5(2s2−3s+4)
2 answers:
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"At least -1" means the number is greater than or equal to -1. (x ≥ -1)
"No more than 9" means the number is less than or equal to 9. (x ≤ 9)
Written together as a compound inequality, these become ...
... -1 ≤ x ≤ 9
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The graph on the number line has solid dots at the ends of the interval, because those values are included in the answer. (If you had -1 < x, for example, you would put an open circle at -1. The circle would be open because -1 is <em>not included</em> in this range.) The dots are joined by a solid line, because all values between (and including) -1 and 9 are part of the answer. (If you had something like x < -1, you would put an open circle at -1 and a solid line to the left with an arrow on the end of it (pointing to the left toward -∞).)
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In general, words used to describe comparisons have their standard English dictionary meanings. ("at most", "at least", "no more than", "no less than", and so on. It can be helpful to consider whether the English meaning includes the "or equal to" case.)
Some folks have trouble with the inequality symbols. The wide-open end is on the side with the larger (more positive) number. "x > 9" means "x is greater than 9", for example. The expression "9 < x" means the same thing, but is read "9 is less than x".
A conventional number line has more positive (greater) numbers on the right. It can be marked any way that is convenient for your problem. In the one shown here, there was a bit of a "plan ahead" issue, as I ran out of space before I got to the 9 at the end of your interval. That is why the break symbol (∥) is there.
Though you can orient the inequality symbols either way (≤ or ≥, < or >), I prefer the "arrow" pointing left (< or ≤). That way, the items in the inequality are in the same order as they are on the number line—smaller on the left. I personally find that less confusing.
