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Rational inequalities Quiz 2

mathematics

 Solve the given equation for the possible range of solutions: $\left| { - 2 - x} \right| \geqslant 9$

 Solve differently for positive value and negative values of $x$ .

$ - 2 - x \leqslant  - 9{\text{ }}or - 2 - x \geqslant 9 \\   - x \leqslant  - 7{\text{ }}or - x \geqslant 11 \\  x \geqslant 7{\text{ }}or \leqslant  - 11 \\  {\text{The solution set is }}(-\infty,-11]\cup [7,\infty )$

 For any positive number $x$, we have $x + \dfrac{1}{x} \geqslant $

 Rearrange the given the inequality to get the minimum value.

${\text{x}} + \dfrac{1}{x} = {\left( {\sqrt x  - \dfrac{1}{{\sqrt x }}} \right)^2} + 2 \\ $  $x > 0$ Then the minimum value of $x$ is $1$. Hence, the value of the surds is $0$ and the minimum value of the inequality is equal to $2$.

 Jack wants to invest $\$ 30,000$. Part of this will be invested in a stable $5\% \,$ simple-interest rate account. The remainder will be "invested" in your father's business, and he says that he'll pay you back with $7\%  $ interest. Your father knows that you're making these investments in order to pay your child's college tuition with the interest income. What is the least you can "invest" with your father, and still $($assuming he really pays you back$)$ get at least $\$ 1900 $ in interest?

 Use the simple interest formula for the amount invested and separately for that of the safe investment and proceed accordingly.

$  x{\text{ is the principal amount, }}I = PRT \\  \left( x \right)\left( {0.07} \right)\left( 1 \right) = 0.07x \\  {\text{The interest on the safe investment will be:}} \\  \left( {30{\text{ }}000{\text{ }}- x} \right)\left( {0.05} \right)\left( 1 \right) = 1500-0.05x \\  {\text{Then the total interest is:}} \\  0.07x  + \left( {1500-0.05x} \right) = 0.02x  + 1500 \\  {\text{We need to get at least }}\$ 1900{\text{, so: }} \\  0.02x + 1500 \geqslant 1900 \\  0.02x \geqslant 400 \\  x \geqslant 20,000 \\  $

 The velocity $v {\text{m/s}}$ of a ball thrown directly up in the air is given by $v = 20 - 10t$, where $t$ is the time in seconds. At what times will the velocity be between $10{\text{ }}m/s$ and $15{\text{ }}m/s$

 Use time velocity relation to form the inequality and solve accordingly.

$v{\text{ is velocity and }}t{\text{ is time}} \\  v = 20 - 10t \\  {\text{Given,}} \\  10 < v < 15 \\  10 < 20 - 10t < 15 \\   - 10 <  - 10t <  - 5 \\  1 > t > 0.5 \\  0.5 < t < 1 \\  $

 Choose the correct option among the following for given expression:  $({a^2}b + {b^2}c + {c^2}a)(c{a^2} + a{b^2} + {c^2}b) \geqslant $ _______ 

 Use Arithmetic Mean and Geometric Mean inequality to solve.

$  {\text{Using AM - GM inequality,}} \\  \left( {{{\text{a}}^2}{\text{b }} + {{\text{b}}^2}{\text{c }} + {{\text{c}}^2}{\text{a}}} \right)\left( {{\text{c}}{{\text{a}}^2} + {\text{a}}{{\text{b}}^2}{\text{  + }}{{\text{c}}^2}{\text{b}}} \right){\text{ }} \geqslant \left( {3\,{{\text{a}}^3}{{\text{b}}^3}{{\text{c}}^3}} \right){\text{ }}\left( {3\,{{\text{a}}^3}{{\text{b}}^3}{{\text{c}}^3}} \right){\text{ }} = {\text{ }}9{{\text{a}}^2}{\text{ }}{{\text{b}}^2}{\text{ }}{{\text{c}}^2} \\  $