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Inequalities and Indirect Reasoning
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13.1 Inequalities
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An inequality is a statement that quantities are not equal. If two quantities are unequal, the first is either greater than or less than the other. The inequality symbols are: , meaning unequal to; , meaning greater than; and , meaning less than. Thus, 4 3 is read four is unequal to three ; 7 2 is read seven is greater than two ; and 1 5 is read one is less than five. Two inequalities may be of the same order or of opposite order. In inequalities of the same order, the same inequality symbol is used; in inequalities of the opposite order, opposite inequality symbols are used. Thus, 5 3 and 10 7 are inequalities of the same order; 5 3 and 7 10 are inequalities of opposite order. Inequalities of the same order may be combined, as follows. The inequalities x y and y z may be combined into x y z, which states that y is greater than x and less than z. The inequalities a b and b c may be combined into a b c, which states b is less than a and greater than c.
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13.1A Inequality Axioms
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Axioms are statements that are accepted as true without proof and are used in the same way as theorems.
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AXIOM 1:
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A quantity may be substituted for its equal in any inequality.
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y and y 10, then x 10.
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Thus if x
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AXIOM 2:
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If the first of three quantities is greater than the second, and the second is greater than the third, then the first is greater than the third.
y and y z, then x z.
Thus if x
AXIOM 3:
The whole is greater than any of its parts.
AM and m/BAD m/BAC in Fig. 13-1.
Thus, AB
Fig. 13-1
13.1B Inequality Axioms of Operation
AXIOM 4:
If equals are added to unequals, the sums are unequal in the same order.
4 and 4 4, we know that 5 4 4 4 (or 9 8). If x 4 5, then x 4 4 5 4 or x 9.
Since 5
AXIOM 5:
Inequalities and Indirect Reasoning
If unequals are added to unequals of the same order, the sums are unequal in the same order.
1, we have 5 4 3 1 (or 9 4). If 2x 4 5 and x 4 8, then 2x 4 x 4 5 8
Since 5 3 and 4 or 3x 13.
AXIOM 6:
If equals are subtracted from unequals, the differences are unequal in the same order.
5 and 3 3, we have 10 3 5 3 (or 7 2). If x 6 9 and 6 6, then x 6 6 9 6 or x 3.
Since 10
AXIOM 7:
If unequals are subtracted from equals, the differences are unequal in the opposite order.
10 and 5 3, we have 10 5 10 3 (or 5 7). If x y 12 and y 5, then x y y 12 5
Since 10 or x 7.
AXIOM 8:
1 Thus if 4x
If unequals are multiplied by the same positive number, the products are unequal in the same
order. 5, then 4s1xd 4 4s5d or x 20.
AXIOM 9:
If unequals are multiplied by the same negative number, the results are unequal in the opposite order.
5, then s 2ds1xd 2 4x 4 s 2ds5d or x 10 or x 10.
Thus if 1x 2
AXIOM 10:
If unequals are divided by the same positive number, the results are unequal in the same order.
20, then 20 or x 4 5.
Thus if 4x
AXIOM 11:
If unequals are divided by the same negative number, the results are unequal in the opposite order.
7x 42, then 7x 7 42 or x 7 6.
Thus if
13.1C
Inequality Postulate
The length of a line segment is the shortest distance between two points.
POSTULATE 1:
13.1D
Triangle Inequality Theorems
The sum of the lengths of two sides of a triangle is greater than the length of the third side. (Corollary: The length of the longest side of a triangle is less than the sum of the lengths of the other two sides and greater than their difference.)
CA AB and AB BC AC.
PRINCIPLE 1:
Thus in Fig. 13-2, BC
Fig. 13-2
PRINCIPLE 2:
In a triangle, the measure of an exterior angle is larger than the measure of either nonadjacent interior angle.
m/BAC and m/BCD m/ABC.
Thus in Fig. 13-2, m/BCD
PRINCIPLE 3:
If the lengths of two sides of a triangle are unequal, the measures of the angles opposite these sides are unequal, the larger angle being opposite the longer side. (Corollary: The largest angle of a triangle is opposite the longest side.)
AC, then m /A m /B.
Thus in Fig. 13-2, if BC
Inequalities and Indirect Reasoning
PRINCIPLE 4:
If the measures of two angles of a triangle are unequal, the lengths of the sides opposite these angles are unequal, the longer side being opposite the larger angle. (Corollary: The longest side of a triangle is opposite the largest angle.)
m/B, then BC AC.
Thus in Fig. 13-2, if m/A
PRINCIPLE 5:
The perpendicular from a point to a line is the shortest segment from the point to the line.
Thus in Fig. 13-3, if PC ' AB and PD is any other line from P to AB, then PC
Fig. 13-3
PRINCIPLE 6:
Fig. 13-4
If two sides of a triangle are congruent to two sides of another triangle, the triangle having the greater included angle has the greater third side.
B C , AC A C , and m/C m/C , then AB AB.
Thus in Fig. 13-4, if BC
PRINCIPLE 7:
If two sides of a triangle are congruent to two sides of another triangle, the triangle having the greater third side has the greater angle opposite this side.
B C , AC A C , and AB A B , then m/C m/C .
Thus in Fig. 13-4, if BC
13.1E
Circle Inequality Theorems
In the same or equal circles, the greater central angle has the greater arc.
m/COD, then mAB mCD.
PRINCIPLE 8:
Thus in Fig. 13-5, if m/AOB
PRINCIPLE 9:
In the same or equal circles, the greater arc has the greater central angle. (This is the converse of Principle 8.)
mCD, then m/AOB m/COD.
Thus in Fig. 13-5, if mAB
Fig. 13-5
PRINCIPLE 10:
Fig. 13-6
In the same or equal circles, the greater chord has the greater minor arc.
CD, then mAB mCD.
Thus in Fig. 13-6, if AB
PRINCIPLE 11:
In the same or equal circles, the greater minor arc has the greater chord. (This is the converse of Principle 10.)
mCD, then AB CD.
Thus in Fig. 13-6, if mAB
PRINCIPLE 12:
In the same or equal circles, the greater chord is at a smaller distance from the center.
CD, then OE OF.
Thus in Fig. 13-7, if AB
Inequalities and Indirect Reasoning
Fig. 13-7
PRINCIPLE 13:
In the same or equal circles, the chord at the smaller distance from the center is the greater chord. (This is the converse of Principle 12.)