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McGraw Hill Integrated II, 2012 View details

6. Inequalities in Two Triangles

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Exercise 7 Page 458

Consider the Hinge Theorem.

5/3< x< 8

Practice makes perfect

For the given triangles, we will find the range of possible values for x.

In order to do that, we will use the Hinge Theorem to compare the side lengths.

Applying the theorem, we can write an inequality for the side lengths. 37^(∘) > 27 ^(∘) ⇒ 2x+3 > 3x-5 Let's solve the inequality for x.

2x+3 >3x-5

AddIneq

LHS+5>RHS+5

2x+8 > 3x

SubIneq

LHS-2x>RHS-2x

8 > x

RearrangeIneq

Rearrange inequality

x < 8

All x-values less than 8 will work. Additionally, notice that 3x-5 and 2x+3 must be greater than 0. cc Inequality II & Inequality III [0.5em] 3x-5 >0 & 2x+3 >0 ⇓ & ⇓ x > 5/3 & x >- 3/2 Since Inequality III contains values that make 3x-5 negative, we can disregard it. Therefore, 53 is the lower limit and 8 is the upper limit of x. Inequality I:&& & x < 8 [0.5em] Inequality II:&& 5/3< & x [0.5em] Combined:&& 5/3<& x< 8

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Exercises p.457-462

Exercise 7 Page 458

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