Find the solution set of the in equation
Question: Find the solution set of the in equation $|2 x-3|1$. Solution: $|2 x-3|1$ Square both sides $\Rightarrow(2 x-3)^{2}1^{2}$ $\Rightarrow 4 x^{2}-12 x+91$ $\Rightarrow 4 x^{2}-12 x+80$ Divide throughout by 4 $\Rightarrow x^{2}-3 x+20$ $\Rightarrow x^{2}-2 x-x+20$ $\Rightarrow x(x-2)-1(x-2)0$ $\Rightarrow(x-1)(x-2)0$ Observe that when $x$ is greater than $2(x-1)(x-2)$ is positive And for each root the sign changes hence We want less than 0 that is negative part Hence $x$ should be between ...
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Question: $\frac{y-(4-3 y)}{2 y-(3 y+4 y)}=\frac{1}{5}$ Solution: Given, $\frac{y-(4-3 y)}{2 y-(3+4 y)}=\frac{1}{5}$ $\Rightarrow$ $5(y-4+3 y)=2 y-3-4 y$ [by cross-multiplication] $\Rightarrow \quad 5(4 y-4)=-3-2 y$ $\Rightarrow$ $20 y-20=-3-2 y$ $\Rightarrow \quad 20 y+2 y=20-3[$ transposing $-20$ to RHS and $-2 y$ to LHS $]$ $\Rightarrow \quad 22 y=17$ $\Rightarrow$ $\frac{22 y}{22}=\frac{17}{22}$ [dividing both sides by 22 ] $\therefore$ $y=\frac{17}{22}$...
Read More →Find the solution set of the in equation
Question: Find the solution set of the in equation $|x-1|2$. Solution: $|x-1|2$ Square both sides $\Rightarrow(x-1)^{2}4$ $\Rightarrow x^{2}-2 x+14$ $\Rightarrow x^{2}-2 x-30$ $\Rightarrow x^{2}-3 x+x-30$ $\Rightarrow x(x-3)+1(x-3)0$ $\Rightarrow(x+1)(x-3)0$ Observe that when $x3(x-3)(x+1)$ is positive And for each root the sign changes hence We want less than 0 that is negative part Hence $x$ should be between $-1$ and 3 for $(x-3)(x+1)$ to be negative Hence $x \in(-1,3)$ Hence solution set for...
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Question: $\frac{0.2 x+5}{3.5 x-3}=\frac{2}{5}$ Solution: Given,$\frac{0.2 x+5}{3.5 x-3}=\frac{2}{5}$ $\Rightarrow$ $5(0.2 x+5)=2(3.5 x-3)$ [by cross-multiplication] $\Rightarrow \quad$ $x+25=7 x-6$ $\Rightarrow$ $x-7 x=-6-25 \quad$ [transposing $7 x$ to LHS and 25 to RHS] $\Rightarrow$ $-6 x=-31$ $\Rightarrow$ $\frac{-6 x}{-6}=\frac{-31}{-6}$ [dividing both sides by $-6$ ] $\therefore$ $x=\frac{31}{6}$...
Read More →Find the solution set of the in equation
Question: Find the solution set of the in equation $\frac{1}{x-2}0$. Solution: $\frac{1}{x-2}0$ We have to find values of $x$ for which $\frac{1}{x-2}$ is less than zero that is negative Now for $\frac{1}{x-2}$ to be negative $x-2$ should be negative that is $x-20$ $\Rightarrow x-20$ $\Rightarrow x2$ Hence $\mathrm{x}$ should be less than 2 for $\frac{1}{x-2}0$ $x2$ means $x$ can take values from $-\infty$ to 2 hence $x \in(-\infty, 2)$ Hence the solution set for $\frac{1}{x-2}0$ is $(-\infty, 2...
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Question: [(5(1 x)) + (3(1 + x))/ (1 2x)] = 8 Solution: We have, [(5(1 x)) + (3(1 + x))/ (1 2x)] = 8 By cross multiplication, we get (5(1 x)) + (3(1 + x)) = 8 (1 2x) 5 5x + 3 + 3x = 8 16 x 8 2x = 8 16x Transposing 8 to RHS it becomes 8 and -16x to LHS it becomes 16x. 16x 2x = 8 8 14x = 0 x = 0/14 x = 0...
Read More →To receive grade A in a course one must obtain an average of 90 marks
Question: To receive grade A in a course one must obtain an average of 90 marks or more in five papers, each of 100 marks. If Tanvy scored 89, 93, 95 and 91 marks in first four papers, find the minimum marks that she must score in the last paper to get grade A in the course. Solution: Let x marks be scored by Tanvy in her last paper. It is given that Tanvy scored 89, 93, 95 and 91 marks in first 4 papers. To receive grade A, she must obtain an average of 90 marks or more. Therefore, $\frac{89+93...
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Question: $\frac{8}{x}=\frac{5}{x-1}$ Solution: Given, $\frac{8}{x}=\frac{5}{x-1}$' $\Rightarrow$ $8(x-1)=5 x$ [by cross-multiplication] $\Rightarrow \quad 8 x-8=5 x$ $\Rightarrow$ $8 x-5 x=8$ [transposing $5 x$ to LHS and $-8$ to RHS] $\Rightarrow \quad 3 x=8$ $\Rightarrow$ $\frac{3 x}{3}=\frac{8}{3}$ [dividing both sides by 3 ] $\therefore$ $x=\frac{8}{3}$...
Read More →How many litres of water will have to be added to 600 litres of the 45%
Question: How many litres of water will have to be added to 600 litres of the 45% solution of acid so that the resulting mixture will contain more than 25%, but less than 30% acid content? Solution: Let x litres of water be added. Then total mixture = x + 600 Amount of acid contained in the resulting mixture is 45% of 600 litres. It is given that the resulting mixture contains more than 25% and less than 30% acid content. Therefore, $45 \%$ of $60025 \%$ of $(x+600)$ And 30% of (x+600) 45% of 60...
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Question: $\frac{2 x-3}{4 x+5}=\frac{1}{3}$ Solution: Given, $\frac{2 x-3}{4 x+5}=\frac{1}{3}$ $\Rightarrow$ $3(2 x-3)=4 x+5$ [by cross-multiplication] $\Rightarrow \quad 6 x-9=4 x+5$ $\Rightarrow$ $6 x-4 x=9+5 \quad$ [transposing $4 x$ to LHS and $-9$ to RHS] $\Rightarrow$ $2 x=14$ $\Rightarrow$ $\frac{2 x}{2}=\frac{14}{2}$ [dividing both sides by 2] $\therefore$ $x=7$...
Read More →A manufacturer has 640 litres of an 8% solution of boric acid.
Question: A manufacturer has 640 litres of an 8% solution of boric acid. How many litres of 2% boric and acid solution be added to it so that the boric acid content in the resulting mixture will be more than 4% but less than 6%. Solution: Let x litres of 2% boric and acid solution be added to 640 litres of 8% solution of boric acid. $\%$ Strength $=\frac{\frac{8}{100} \times 640+\frac{2}{100} \times x}{640+x}$ $=\frac{5120+2 x}{100(640+x)}$ It is given that boric acid content in the resulting mi...
Read More →The watering acidity in a pool is considered normal when the average
Question: The watering acidity in a pool is considered normal when the average pH reading of three daily measurements is between 8.2 and 8.5. If the first two pH reading 8.48 and 8.35, find the range of the pH values for the third reading that will result in the acidity level is normal. Solution: Let x be the third pH value. Now, it is given that the average pH reading of three daily measurements is between 8.2 and 8.5 Also, the first two pH readings are 8.48 and 8.35 Therefore, $8.2\frac{8.48+8...
Read More →A company manufactures cassettes.
Question: A company manufactures cassettes. Its cost and revenue function are C(x) = 25000 + 30x and R(x) = 43x respectively, where x is the number of cassettes produced and sold in a week. How many cassettes must be sold by the company to realize some profit? Solution: Given: Cost function C(x) = 25000 + 30x Revenue function R(x) = 43x To Find: Number of cassettes to be sold to realize some profit In order, to gain profit: R(x) C(x) Therefore, 43x 25000 + 30x 25000 + 30x 43x Subtracting 30x fro...
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Question: $\frac{5 x}{2 x-1}=2$ Solution: Given,$\frac{5 x}{2 x-1}=2$ $5 x=2(2 x-1)$ [by cross-multiplication] $\Rightarrow \quad 5 x=4 x-2$ $\Rightarrow$ $5 x-4 x=-2$ [transposing $4 x$ to LHS] $\therefore \quad x=-2$...
Read More →Find all pairs of consecutive even positive integers both of
Question: Find all pairs of consecutive even positive integers both of which are larger than 8 such that their sum is less than 25. Solution: Let the pair of consecutive even positive integers be x and x + 2. So, it is given that both the integers are greater than 8 Therefore, x 8 and x + 2 8 When x + 2 8 Subtracting 2 from both the sides in above equation $x+2-28-2$ $x6$ Since x 8 and x 6 Therefore, x 8 It is also given that sum of both the integers is less than 25 Therefore $x+(x+2)25$ $x+x+22...
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Question: $\frac{3 x-8}{2 x}=1$ Solution: Given,$\frac{3 x-8}{2 x}=1$ $\Rightarrow$ $3 x-8=2 x$ [bycross-multiplication] $\Rightarrow \quad 3 x-2 x=8 \quad$ [transposing $2 x$ to LHS and 8 to RHS] $\therefore$ $x=8$...
Read More →Find all pairs of consecutive even positive integers both of
Question: Find all pairs of consecutive even positive integers both of which are larger than 8 such that their sum is less than 25. Solution: Let the pair of consecutive even positive integers be x and x + 2 So, it is given that both the integers are greater than 8 Therefore $x8$ and $x+28$ When, x + 2 8 Subtracting 2 from both the sides in above equation $x+2-28-2$ $x6$ Since $x8$ and $x6$ Therefore x 8 It is also given that sum of both the integers is less than 25 Therefore, $x+(x+2)25$ $x+x+2...
Read More →Two numbers differ by 40.
Question: Two numbers differ by 40. When each number is increased by 8, the bigger becomes thrice the lesser number. If one number is x, then the other number is (40 x). Solution: False Given, one number $=x$ and other number $=40-x$ Let $(40-x)x$ Then, according to the question, $40-x+8=3(x+8)$ $\Rightarrow \quad 48-x=3 x+24$ $\Rightarrow \quad-x-3 x=24-48 \quad$ [transposing $3 x$ to LHS and 48 to RHS] $\Rightarrow$ $-4 x=-24$ $\Rightarrow$ $x=-24 \times\left(-\frac{1}{4}\right)$ $\therefore$ ...
Read More →If the sum of two consecutive numbers
Question: If the sum of two consecutive numbers is 93 and one of them is x, then the other number is 93 x. Solution: True Given, one of the consecutive number = x Then, the next consecutive number = x + 1 According to the question, $x+x+1=93$ $\Rightarrow$ $2 x=93-1$ [transposing 1 to RHS] $\Rightarrow \quad 2 x=92$ $\Rightarrow$ $\frac{2 x}{2}=\frac{92}{2}$ [dividing both sides by 2] $\therefore$$x=46$ Hence, the other consecutive number $=46+1=47=93-46=93-x$...
Read More →Solve the following systems of linear in equations:
Question: Solve the following systems of linear in equations: $1 \leq|x-2| \leq 3$ Solution: $1 \leq|x-2|$ and $|x-2| \leq 3$ When, $|x-2| \geq 1$ Then $x-2 \leq-1$ and $x-2 \geq 1$$x-2 \leq-1$ and $x-2 \geq 1$ Now when, $x-2 \leq-1$ Adding 2 to both the sides in above equation $x-2+2 \leq-1+2$ $x \leq 1$ Now when, $x-2 \geq 1$ Adding 2 to both the sides in above equation $x-2+2 \geq 1+2$ $x \geq 3$ For $|x-2| \geq 1: x \leq 1$ or $x \geq 3$ When, $|x-2| \leq 3$ Then, $x-2 \geq-3$ and $x-2 \leq ...
Read More →If x is an even number,
Question: If x is an even number, then the next even number is 2(x +1). Solution: False Given, x is an even number. Then, the next even number is (x + 2)....
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Question: If $\frac{x}{11}$, then $\mathrm{x}=\frac{11}{15}$ Solution: False Given, $\frac{x}{11}=15$ $\Rightarrow$ $x=11 \times 15$ [by cross-multiplication]...
Read More →If 6x = 18,
Question: If 6x = 18, then 18x = 54. Solution: True Given, $6 x=18$ $\Rightarrow$ $3 \times 6 x=18 \times 3$ [multiplying both sides by 3 ] $\Rightarrow \quad 18 x=54$...
Read More →Solve the following systems of linear in equations:
Question: Solve the following systems of linear in equations: $-124-\frac{3 x}{-5} \leq 2$ Solution: $-124-\frac{3 x}{-5}$ and $4-\frac{3 x}{-5} \leq 2$ When, $-124-\frac{3 x}{-5}$ $4-\frac{3 x}{-5}-12$ Subtracting 4 from both the sides in above equation $4-\frac{3 x}{-5}-412-4$ $-\frac{\frac{3 x}{-5}} -16$ $\frac{3 x}{5}-16$ Multiplying both the sides by 5 in the above equation $\left(\frac{3 x}{5}\right)(5)-16(5)$ $3 x-80$ Dividing both the sides by 3 in above equation $\left(\frac{3 x}{3}\rig...
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Question: If $\frac{x}{3}+1=\frac{7}{15}$, then $\frac{x}{3}=\frac{6}{15}$ Solution: False Given, $\frac{x}{3}+1=\frac{7}{15}$ $\Rightarrow$ $\frac{x}{3}=\frac{7}{15}-1$ [transposing 1 to RHS] $\Rightarrow$ $\frac{x}{3}=\frac{7-15}{15}$ $\Rightarrow$ $\frac{x}{3}=\frac{-8}{15}$...
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