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20. Write an inequality to describe the possible values of x. Co 11 2x19 dignel or bar [ 14 2x - 5 12 415 14 12 38°

20. Write an inequality to describe the possible values of x. Co 11 2x19 dignel or bar [ 14 2x - 5 12 415 14 12 38°

Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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Write an inequality to describe the possible values of x. 

**20. Write an inequality to describe the possible values of x.** There are two diagrams of triangles shown: **Left Triangle:** - The sides of the triangle measure 12 units, 14 units, and a side labeled as \(2x - 5\). - The internal angle opposite the \(2x - 5\) side is \(41^\circ\). **Right Triangle:** - The sides of the triangle measure 11 units, 12 units, and 14 units. - The internal angle opposite the 12-unit side is \(38^\circ\). To find the possible values of \(x\), we need to set up an inequality that reflects the relationship imposed by the geometric properties of the triangles and their angles. 1. Since the side \(2x - 5\) should form a valid triangle with sides 12 and 14, the value of \(2x - 5\) must be less than the sum of the other two sides (12 + 14) and greater than the absolute difference between the other two sides (\(|12 - 14|\)). Therefore: \[|12 - 14| < 2x - 5 < 12 + 14\] \[2 < 2x - 5 < 26\] 2. Solving these inequalities for \(x\): For the left part of the inequality: \[ 2 < 2x - 5 \] Adding 5 to both sides: \[ 7 < 2x \] Dividing by 2: \[ 3.5 < x \] For the right part of the inequality: \[ 2x - 5 < 26 \] Adding 5 to both sides: \[ 2x < 31 \] Dividing by 2: \[ x < 15.5 \] Thus, combining both parts: \[ 3.5 < x < 15.5 \] Therefore, the inequality that describes the possible values of \(x\) is: \[ 3.5 < x < 15.5 \]
Transcribed Image Text:**20. Write an inequality to describe the possible values of x.** There are two diagrams of triangles shown: **Left Triangle:** - The sides of the triangle measure 12 units, 14 units, and a side labeled as \(2x - 5\). - The internal angle opposite the \(2x - 5\) side is \(41^\circ\). **Right Triangle:** - The sides of the triangle measure 11 units, 12 units, and 14 units. - The internal angle opposite the 12-unit side is \(38^\circ\). To find the possible values of \(x\), we need to set up an inequality that reflects the relationship imposed by the geometric properties of the triangles and their angles. 1. Since the side \(2x - 5\) should form a valid triangle with sides 12 and 14, the value of \(2x - 5\) must be less than the sum of the other two sides (12 + 14) and greater than the absolute difference between the other two sides (\(|12 - 14|\)). Therefore: \[|12 - 14| < 2x - 5 < 12 + 14\] \[2 < 2x - 5 < 26\] 2. Solving these inequalities for \(x\): For the left part of the inequality: \[ 2 < 2x - 5 \] Adding 5 to both sides: \[ 7 < 2x \] Dividing by 2: \[ 3.5 < x \] For the right part of the inequality: \[ 2x - 5 < 26 \] Adding 5 to both sides: \[ 2x < 31 \] Dividing by 2: \[ x < 15.5 \] Thus, combining both parts: \[ 3.5 < x < 15.5 \] Therefore, the inequality that describes the possible values of \(x\) is: \[ 3.5 < x < 15.5 \]
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