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ssrs barcode font free Inequalities and Indirect Reasoning in ObjectiveC
Inequalities and Indirect Reasoning QR Code Decoder In ObjectiveC Using Barcode Control SDK for iPhone Control to generate, create, read, scan barcode image in iPhone applications. QR Code ISO/IEC18004 Generation In ObjectiveC Using Barcode creation for iPhone Control to generate, create QR Code JIS X 0510 image in iPhone applications. 13.1 Inequalities
Quick Response Code Recognizer In ObjectiveC Using Barcode decoder for iPhone Control to read, scan read, scan image in iPhone applications. Bar Code Maker In ObjectiveC Using Barcode creation for iPhone Control to generate, create barcode image in iPhone applications. 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. QR Code Maker In C# Using Barcode printer for .NET framework Control to generate, create QR Code ISO/IEC18004 image in Visual Studio .NET applications. Encoding QRCode In Visual Studio .NET Using Barcode generation for ASP.NET Control to generate, create QR Code ISO/IEC18004 image in ASP.NET applications. 13.1A Inequality Axioms
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Thus if x
AXIOM 3: The whole is greater than any of its parts.
AM and m/BAD m/BAC in Fig. 131. Thus, AB
Fig. 131 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. 132, BC
Fig. 132 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. 132, 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. 132, 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. 132, 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. 133, if PC ' AB and PD is any other line from P to AB, then PC
Fig. 133 PRINCIPLE 6: Fig. 134 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. 134, 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. 134, 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. 135, 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. 135, if mAB
Fig. 135 PRINCIPLE 10: Fig. 136 In the same or equal circles, the greater chord has the greater minor arc.
CD, then mAB mCD.
Thus in Fig. 136, 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. 136, 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. 137, if AB
Inequalities and Indirect Reasoning
Fig. 137 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.)

