Lesson 4
Square Roots on the Number Line
Problem 1
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Find the exact length of each line segment.
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Estimate the length of each line segment to the nearest tenth of a unit. Explain your reasoning.
Solution
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Problem 2
Plot each number on the \(x\)-axis: \(\sqrt{16},\text{ } \sqrt{35},\text{ } \sqrt{66}\). Consider using the grid to help.
![quadrant 1, x axis, 0 to 10, by 1's. y axis, 0 to 8, by 1's.](https://cms-im.s3.amazonaws.com/x9jTyGudkLV1vswPXUV3MT9e?response-content-disposition=inline%3B%20filename%3D%228-8.8.B.4.revised.image.06.taller.png%22%3B%20filename%2A%3DUTF-8%27%278-8.8.B.4.revised.image.06.taller.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240630%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240630T181455Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=144f5164e73fce97b4e2d691de5643753265f7762ece90db8cb4b483b7b37292)
Solution
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Problem 3
Use the fact that \(\sqrt{7}\) is a solution to the equation \(x^2 = 7\) to find a decimal approximation of \(\sqrt{7}\) whose square is between 6.9 and 7.1.
Solution
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Problem 4
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Explain how you know that \(\sqrt{37}\) is a little more than 6.
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Explain how you know that \(\sqrt{95}\) is a little less than 10.
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Explain how you know that \(\sqrt{30}\) is between 5 and 6.
Solution
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Problem 5
Plot each number on the number line: \(\displaystyle 6, \sqrt{83}, \sqrt{40}, \sqrt{64}, 7.5\)
![A number line with 6 evenly spaced tick marks and the integers 5 through 10 are indicated.](https://cms-im.s3.amazonaws.com/rqm9geeGdG7Dy1VGX3i6PLYT?response-content-disposition=inline%3B%20filename%3D%228-8.8.B.PP.Image.07.png%22%3B%20filename%2A%3DUTF-8%27%278-8.8.B.PP.Image.07.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240630%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240630T181455Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=058438844ae7d7e8ca2a9a320573498059cd1c97467a82e8595b606fefcba05f)
Solution
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Problem 6
The equation \(x^2=25\) has two solutions. This is because both \(5 \boldcdot 5 = 25\), and also \(\text-5 \boldcdot \text-5 = 25\). So, 5 is a solution, and also -5 is a solution.
Select all the equations that have a solution of -4:
\(10+x=6\)
\(10-x=6\)
\(\text-3x=\text-12\)
\(\text-3x=12\)
\(8=x^2\)
\(x^2=16\)
Solution
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Problem 7
Find all the solutions to each equation.
- \(x^2=81\)
- \(x^2=100\)
- \(\sqrt{x}=12\)
Solution
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Problem 8
The points \((12, 23)\) and \((14, 45)\) lie on a line. What is the slope of the line?
Solution
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(From Unit 5, Lesson 4.)