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Elementary Algebra

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question 1

Solve the equation for $x$, and write your answer in simplified form$$\frac{5}{12}\left(24x-36\right)=-\frac{1}{8}\left(32-16x\right)$$


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question 2

$\def\Va{2}\def\VPlusa{+2}\def\VMinusa{-2}\def\Vb{14}\def\VPlusb{+14}\def\VMinusb{-14}\def\Vc{1}\def\VPlusc{+1}\def\VMinusc{-1}\def\Vd{-7}\def\VPlusd{-7}\def\VMinusd{+7}\def\Ve{-8}\def\VPluse{-8}\def\VMinuse{+8}\def\Vf{-9}\def\VPlusf{-9}\def\VMinusf{+9}$

Solve fro $x$.$$\Ve x\VPlusa >\Vb $$

$\Vf x\VPlusc <\Vd $


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question 3

$\def\VsSubOneSub{52500}\def\VPlussSubOneSub{+52500}\def\VMinussSubOneSub{-52500}\def\VsSubTwoSub{60000}\def\VPlussSubTwoSub{+60000}\def\VMinussSubTwoSub{-60000}\def\Vd{s_{2}-s_{1}}\def\VPlusd{+s_{2}-s_{1}}\def\VMinusd{-s_{2}-s_{1}}$

John's yearly salary was 52,500 last year. This year he received a raise and his salary was increased to 60,000. What is the percent change?

Round to the nearest tenth of a percent.


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question 4

Henry is mixing pistachios and almonds to make $12.5$ pounds of trail mix. Pistachios cost $\$16$ per pound and almonds cost $\$6$ per pound. How many pounds $x$ of pistachios and how many pounds of almonds should Henry use for the trail mix to cost $\$10$ per pound?


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question 5

Sean was paid $\$50000$ for a cell phone app that he wrote and wants to invest it to save for his daughter's education. He wants to put some of the money into a bond that pays $4\%$ annual interest and the rest into stocks that pay $9\%$ annual interest. If he wants to earn $7.4\%$ annual interest on the total amount, how much money should he invest in each account?

Let $x$ be the amount invested in the $4\%$ bond.


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question 6

Find the slope of the line shown below. 


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question 7

$\def\Vm{-4}\def\VPlusm{-4}\def\VMinusm{+4}\def\VxSubOneSub{-6}\def\VPlusxSubOneSub{-6}\def\VMinusxSubOneSub{+6}\def\VySubOneSub{8}\def\VPlusySubOneSub{+8}\def\VMinusySubOneSub{-8}$

Find an equation of a line with slope $m=-4$  that contains the point $\left(-6,8\right)$. Write the equation in slope–intercept form.


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question 8

A cup of coffee and two eggs cost $\$10.50$. Two cups of coffee and three eggs cost $\$18.00$. How much does one coffee and one egg cost on their own?

Let the cost of a cup of coffee be $c$ and the cost of an egg be $e$.


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question 9

$\def\Vy{\frac{4}{3}x+1}\def\VPlusy{+\frac{4}{3}x+1}\def\VMinusy{-\frac{4}{3}x+1}$

Solve the following system of equations in two variables.
\begin{align*} {4x-3y}&={10}\\ {y\,}&={\frac{4}{3}x+1} \end{align*}


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question 10

There are $300$ people at a wedding. The number of children is two-thirds of the number of adults at the wedding. Find the number of children and adults at the wedding.


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question 11

$\def\Va{-5}\def\VPlusa{-5}\def\VMinusa{+5}\def\Vb{9}\def\VPlusb{+9}\def\VMinusb{-9}\def\Vc{2}\def\VPlusc{+2}\def\VMinusc{-2}\def\Vd{19}\def\VPlusd{+19}\def\VMinusd{-19}\def\Ve{-2}\def\VPluse{-2}\def\VMinuse{+2}\def\Vf{-9}\def\VPlusf{-9}\def\VMinusf{+9}\def\Vy{\left(a+d\right)q+\left(b+e\right)m+\left(c+f\right)z}\def\VPlusy{+\left(a+d\right)q+\left(b+e\right)m+\left(c+f\right)z}\def\VMinusy{-\left(a+d\right)q+\left(b+e\right)m+\left(c+f\right)z}$

Simplify $$y=\left(\Va q\VPlusb m\VPlusc z\right)+\left(\Vd q\VPluse m\VPlusf z\right)$$


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question 12

$\def\Va{12}\def\VPlusa{+12}\def\VMinusa{-12}\def\Vb{9}\def\VPlusb{+9}\def\VMinusb{-9}\def\Vc{2}\def\VPlusc{+2}\def\VMinusc{-2}\def\Vd{11}\def\VPlusd{+11}\def\VMinusd{-11}\def\Ve{60}\def\VPluse{+60}\def\VMinuse{-60}\def\VRx{12x+9}\def\VPlusRx{+12x+9}\def\VMinusRx{-12x+9}\def\VCx{2x+11}\def\VPlusCx{+2x+11}\def\VMinusCx{-2x+11}\def\VPx{R\left(x\right)-C\left(x\right)}\def\VPlusPx{+R\left(x\right)-C\left(x\right)}\def\VMinusPx{-R\left(x\right)-C\left(x\right)}$

Carlos owns a business. He determines that he can model his  revenue $R\left(x\right)\,=\,\Va x\VPlusb $ and cost $C\left(x\right)\,=\,\Vc x\VPlusd $ where $x$ is the amount of units sold.

Given these functions, determine the net profit $P\left(x\right)$ that Carlos is making as a function of $x$. If Carlos sells $\Ve $ units, how much profit will he make?


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question 13

$\def\Va{7}\def\VPlusa{+7}\def\VMinusa{-7}\def\Vb{18}\def\VPlusb{+18}\def\VMinusb{-18}\def\Vm{25}\def\VPlusm{+25}\def\VMinusm{-25}\def\Vn{126}\def\VPlusn{+126}\def\VMinusn{-126}\def\Vq{x+b}\def\VPlusq{+x+b}\def\VMinusq{-x+b}$

Given the quadratic $x^{2}\VPlusm x\VPlusn $ along with the partially factored form $\left(x\VPlusa \right)\left(x+b\right)$, find the missing number $b$ in the second factor.


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question 14

$\def\Vc{-5}\def\VPlusc{-5}\def\VMinusc{+5}\def\Vd{9}\def\VPlusd{+9}\def\VMinusd{-9}\def\Vy{e\left(x^{2}+mx+n\right)}\def\VPlusy{+e\left(x^{2}+mx+n\right)}\def\VMinusy{-e\left(x^{2}+mx+n\right)}\def\Ve{4}\def\VPluse{+4}\def\VMinuse{-4}\def\Vm{4}\def\VPlusm{+4}\def\VMinusm{-4}\def\Vn{-45}\def\VPlusn{-45}\def\VMinusn{+45}\def\VbSubOneSub{16}\def\VPlusbSubOneSub{+16}\def\VMinusbSubOneSub{-16}\def\VcSubOneSub{-180}\def\VPluscSubOneSub{-180}\def\VMinuscSubOneSub{+180}$

Factor the quadratic 

$$\Ve x^{2}\VPlusbSubOneSub x\VPluscSubOneSub $$


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question 15

$\def\Va{-2}\def\VPlusa{-2}\def\VMinusa{+2}\def\VbSubOneSub{18}\def\VPlusbSubOneSub{+18}\def\VMinusbSubOneSub{-18}\def\Vc{-5}\def\VPlusc{-5}\def\VMinusc{+5}\def\Vd{-2}\def\VPlusd{-2}\def\VMinusd{+2}\def\Ve{10}\def\VPluse{+10}\def\VMinuse{-10}\def\Vm{-86}\def\VPlusm{-86}\def\VMinusm{+86}\def\Vn{-36}\def\VPlusn{-36}\def\VMinusn{+36}$

Given the quadratic $\Ve x^{2}\VPlusm x\VPlusn $ along with the factored form $\;\left(\Vc x\VPlusa \right)\left(\Vd x+b\right)$, find the missing constant $b$.


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question 16

$\def\Va{-18}\def\VPlusa{-18}\def\VMinusa{+18}\def\Vb{-19}\def\VPlusb{-19}\def\VMinusb{+19}\def\Vc{7}\def\VPlusc{+7}\def\VMinusc{-7}\def\Vy{\frac{-18x}{x-7}+\frac{-19}{x-7}}\def\VPlusy{+\frac{-18x}{x-7}+\frac{-19}{x-7}}\def\VMinusy{-\frac{-18x}{x-7}+\frac{-19}{x-7}}$

Evaluate

$$\Vy $$


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question 17

Add and simlify$$\frac{7}{3xy+y^{2}}+\frac{4}{9x^{2}-y^{2}}\,$$


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question 18

Solve $${\sqrt[\,]{3x-3}}+2={\sqrt[\,\,]{6x+1}}$$


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question 19

Solve the radical equation
$$2{\sqrt[]{r+\frac{29}{4}}}=r+6$$


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question 20

$\def\Vb{\frac{3}{2}}\def\VPlusb{+\frac{3}{2}}\def\VMinusb{-\frac{3}{2}}\def\Vc{\left(\frac{1}{2}b\right)^{2}}\def\VPlusc{+\left(\frac{1}{2}b\right)^{2}}\def\VMinusc{-\left(\frac{1}{2}b\right)^{2}}$

Determine the value of $c$ that makes  $m^{2}\VPlusb m+c$ a perfect square. Then write the result as a binomial squared.


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question 21

$\def\Vn{3}\def\VPlusn{+3}\def\VMinusn{-3}\def\VmSubOneSub{4}\def\VPlusmSubOneSub{+4}\def\VMinusmSubOneSub{-4}\def\Vm{24}\def\VPlusm{+24}\def\VMinusm{-24}\def\Vp{9}\def\VPlusp{+9}\def\VMinusp{-9}$

Determine the values of $a$, $b$, and $c$ so that $\Vn x^{2}\VPlusm x\VPlusp =a\left(x-b\right)^{2}+c$


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