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This is one of our many quizzes on Graph real numbers on a number line. Click on this to access more such competitive quizzes on Real Numbers.

Graph real numbers on a number line Quiz 2

mathematics

 Solve the inequality $-2\left( -13x+1\dfrac{3}{4} \right)\le 309\dfrac{1}{2}$ and plot it on the number line. 

 Open the brackets and then rearrange the terms such that only $x$ remains in the LHS. Then, plot the value of $x$.

  $-2\left( -13x+1\dfrac{3}{4} \right)\le 309\dfrac{1}{2} \\ \Rightarrow 26x-\dfrac{14}{4}\le \dfrac{619}{2} \\ \Rightarrow x\le 12.03 \\ \text{So, we can plot it as below,} \\ $<img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/cf188d76-4227-437e-9e1a-f638d73d85857278098909140371684.png"/>

 How will the plot of $0.5\left( 2.55-3.25e \right) > 5\left( \left\{ 2e+2 \right\}+0.5 \right)$ on the number line look like?

 Simplify both sides of the inequality and then rearrange the terms to get $e$ on the RHS. Then, make a plot of the obtained range of $e$.

  $ 0.5\left( 2.55-3.25e \right) > 5\left( \left\{ 2e+2 \right\}+0.5 \right) \\ \Rightarrow 1.275-1.675e > 10e+12.5 \\ \Rightarrow -11.225 > 11.675e \\ \Rightarrow -0.96 > e \\ \text{Since the value of “e” is less than -0.96} \\ \text{We can plot it as below,} \\ $<img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/90c03334-fd0d-4fc2-8336-21daa6d746d47254255406304017973.png"/>

 Plot $\dfrac{1}{2}\left( 4r+8 \right)\le r+1 < \dfrac{1}{4}\left( 20+8r \right)$ on the number line. What will be the interval notation?

 Consider both parts of the inequality separately, solve for $r$ and then combine them to get the possible values of $r$.

 $\dfrac{1}{2}\left( 4r+8 \right)\le r+1 < \dfrac{1}{4}\left( 20+8r \right) \\ \Rightarrow 2r+4\le r+1 < 5+2r \\ \text{Considering separately, we have} \\ 2r+4\le r+1 \\ \Rightarrow r\le -3 \\ r+1 < 5+2r \\ \Rightarrow -4 < r \\ \text{Combining, we get }-4 < r\le -3\text{ and the plot is as below,} \\ $ <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/e4d33b97-5ee6-49ee-b555-b46bddb56ae95716772967072454648.png"/>

 Plot the following compound inequality on the number line.<br>$\sqrt{{{x}^{2}}+27} < -2x$  or $x+5 > x\left( x+1 \right)$ 

 The values of $x$ for both cases must be found out and ‘or’ suggests that $x$ can have either of the values from both cases.

 $  \text{Considering first one, we get} \\ \sqrt{{{x}^{2}}+27} < -2x \\ \Rightarrow {{x}^{2}}+27 < 4{{x}^{2}} \\ \Rightarrow 27 < 3{{x}^{2}} \\ \Rightarrow 3 < x \\ \text{Considering second one, we get} \\  x+5 > x\left( x+1 \right) \\ \Rightarrow 5 > {{x}^{2}} \\ \Rightarrow 2.23 > x \\\text{Now, plotting the compound inequality, we have} \\ $<img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/a9c5fb9a-75c0-4670-95e2-66b9d85dd9d11035595298876595729.png"/>

 Amira has entered a puzzle competition. In the first round, she scores 20 points. In the second round, she scores 5 more than twice the score in the first round. She completes the final round and her total score is less than 100. If she had got 150 points or more, she would have won the competition. For what range of scores in the final round, would she have failed the competition?

 Form two inequalities with the given conditions and obtain values of $x$. Then, plot them on the number line and find the possible values.

 $ \text{Let Amira’s score in the final round be $x$. } \\ \text{Since the total score was less than 100,} \\ \text{Then the inequality is} \\ 20+\left( 5+2\left( 20 \right) \right)+x < 100 \\ \Rightarrow 65+x < 100 \\ \Rightarrow x < 35 \\ \text{She would have won if } 65+x\ge 150 \\ \Rightarrow x\ge 85 \\ \text{So, the plot would be }\\ $ <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/20a4c949-0edc-4797-ba29-c75a66362a686677299482769684808.png"/>$\\ \text{It is clear that if she had scored between 35 and 85, she would have failed.} \\ \text{So, the range of values are }35 < x < 85 $.