There are 264 students going on a field trip to the city. If the school is only taking 8 buses, what is the total number of stud
2 answers:
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Answer:
the comparison must be reversed when multiplying by a negative
Step-by-step explanation:
The rules of equality apply to solving inequalities, with the exception that multiplication or division by a negative number reverses the sense of the comparison:
-x > 1
x < -1 . . . . . multiply both sides by -1
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Effectively, multiplication (or division) by a negative number is equivalent to reflection across the origin. Things that were ordered left/right (on the number line) are ordered right/left after such a reflection:
-2 < -1
1 < 2
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<em>Additional comment</em>
Application of any function requires that you pay attention to ordering. Some functions naturally reverse the order; others do so only on specific domains.
Consider f(x) = 1/x.
1 < 2
f(1) > f(2) . . . . because the slope of the f(x) function is negative everywhere.
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<em>In the first and second quadrants</em>, the cosine function also reverses order.
20° > 10°
cos(20°) < cos(10°)
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Probably the most commonly encountered function used with inequalities is the absolute value function.
|x-3| > 2
This function has one domain where the slope is negative, and another domain where the slope is positive.
For x < 3, the function negates its argument, so we have ...
-(x -3) > 2
-x +3 > 2
-x > -1
x < 1 . . . . . everywhere consistent with x < 3
For x ≥ 3, the function does nothing, so we have ...
x -3 > 2
x > 5
The solution to this absolute value inequality is (x < 1) ∪ (x > 5).
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You can also resolve negative coefficients by adding the opposite. Addition and subtraction never require any change to the comparison operator.
-x > 1
0 > x +1 . . . . x is added
-1 > x . . . . . . -1 is added
<u>Part a)</u> Use a system of inequalities to model the scenario
Let
x-------> represent babysitting hours
y-------> represent tutoring hours
we know that
--------> inequality
--------> inequality
<u>Part b)</u> Use the model created in part A to create a graph representing Jeanne’s probable income earned and possible number of hours worked this week
we know that
--------> inequality
This inequality represent Jeanne’s possible number of hours worked this week
Using a graph tool
the solution of this inequality is the shaded area
see the attached figure N
--------> inequality
This inequality represent Jeanne’s probable income earned this week
Using a graph tool
the solution of this inequality is the shaded area
see the attached figure N
<u>Part c)</u> Analyze the set of coordinate values that represent solutions for the model created in part A. Choose one of the coordinates within the solution and algebraically prove that the coordinate represents a true solution for the model
we have the system of inequalities
--------> inequality
--------> inequality
the solution of the system of inequalities is the shaded area
using a graph tool
see the attached figure N
Let
------> see the attached figure N
The point belongs to the shaded area, so it is a system solution.
If the point is a solution, it must satisfy both inequalities.
<u>Prove algebraically </u>
substitute the value of x and y in the inequalities
<u>inequality </u>
------> is ok
<u>inequality </u>
-------> is ok
the point A satisfy both inequalities
therefore
the point A is a solution of the system
