QuestionFind the values of $x$ , $BC$ , $CD$ , and $BD$ given $BD=7x-10$ , $BC=4x-29$ , and $CD=5x-9$ .

Studdy Solution

STEP 1

Assumptions1. The given equations are $BD=7x-10$ , $BC=4x-29$ , and $CD=5x-9$ . We are assuming that $BD$ , $BC$ , and $CD$ are the sides of a triangle, and the sum of any two sides of a triangle is greater than the third side.

STEP 2

The sum of the lengths of two sides of a triangle is greater than the length of the third side. So, $BC + CD > BD$ and $BC + BD > CD$ and $BD + CD > BC$ . We can write these inequalities as $4x -29 +5x -9 >7x -10$ $4x -29 +7x -10 >5x -9$ $7x -10 +5x -9 >4x -29$

STEP 3

implify the inequalities $9x -38 >7x -10$ $11x -39 >5x -9$ $12x -19 >x -29$

STEP 4

Rearrange the inequalities to solve for $x$ $9x -7x >38 +10$ $11x -x >39 +9$ $12x -4x >19 +29$

STEP 5

implify the inequalities $2x >48$ $x >48$ $8x >48$

STEP 6

Divide each inequality by the coefficient of $x$ to solve for $x$ $x >48 /2$ $x >48 /6$ $x >48 /8$

STEP 7

Calculate the values of $x$ $x >24$ $x >$ $x >6$

STEP 8

The largest value that $x$ must be greater than is $24$ . Hence, $x$ must be greater than $24$ .

STEP 9

Now, we can find the values of $BC$ , $CD$ , and $BD$ by substituting $x >24$ into the original equations.

STEP 10

Substitute $x >24$ into the equation for $BC$ $BC =4x -29 >4(24) -29$

STEP 11

Calculate the value of $BC$ $BC >96 -29$ $BC >67$

STEP 12

Substitute $x >24$ into the equation for $CD$ $CD =5x -9 >5(24) -9$

STEP 13

Calculate the value of $CD$ $CD >120 -9$ $CD >111$

STEP 14

Substitute $x >24$ into the equation for $BD$ $BD =7x -10 >7(24) -10$

STEP 15

Calculate the value of $BD$ $BD >168 -10$ $BD >158$ The values are $x >24$ $BC >67$ $CD >111$ $BD >158$

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