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Big Ideas Math Geometry, 2014

BI

Big Ideas Math Geometry, 2014 View details

5. Indirect Proof and Inequalities in One Triangle

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Exercise 41 Page 342

Consider the Triangle Inequality Theorem.

2 17

Practice makes perfect

Let's analyze the given triangle.

According to the Triangle Inequality Theorem, the sum of any two sides in our triangle must be greater than the third side. We can use this fact to create three inequalities with the given expressions. rc (I): & (2x+3)+(3x-1)>6x-11 (II): & (2x+3)+(6x-11)>3x-1 (III): & (3x-1)+(6x-11)>2x+3Let's start by solving Inequality (I).

(2x+3)+(3x-1)>6x-11

▼

Solve for x

Remove parentheses

2x+3+3x-1>6x-11

Add and subtract terms

5x+2>6x-11

LHS-5x=RHS-5x

2>x-11

LHS+11=RHS+11

13>x

Rearrange inequality

x<13

We will continue with Inequality (II).

(2x+3)+(6x-11)>3x-1

▼

Solve for x

Remove parentheses

2x+3+6x-11>3x-1

Add and subtract terms

8x-8>3x-1

LHS-3x=RHS-3x

5x-8>-1

LHS+8=RHS+8

5x>7

.LHS /6.=.RHS /6.

x>7/5

Write fraction as a mixed number

x>1 25

Finally, we will solve for x in Inequality (III).

(3x-1)+(6x-11)>2x+3

▼

Solve for x

Remove parentheses

3x-1+6x-11>2x+3

Add and subtract terms

9x-12>2x+3

LHS-2x=RHS-2x

7x-12>3

LHS+12=RHS+12

7x>15

.LHS /7.=.RHS /7.

x>15/7

Write fraction as a mixed number

x>2 17

Now that we have all of the possible limitations for the value of x, we need to find where the three inequalities overlap.

In interval notation, this can be written as the following compound inequality. 2 17

Subchapter links

Communicate Your Answer p.335

Monitoring Progress p.336-339