Full Length THEA Mathematics Practice Test

The correct answer is \(20\)
Add \(8\) both sides of the equation \( 8\ x \ - \ 8=24\) gives \(8 \ x =24 \ + \ 8=32\).
Dividing each side of the equation \( 8 \ x=32\) by \(8\) gives \(x=4\).
Substituting \(4\) for \(x\) in the expression \(6\ x \ - \ 4\) gives \(6\ (4)\ - \ 4=20\).

The correct answer is \(( 4 , -\ 4 )\)
Method 1: Plug in the values of \( x \) and \( y \) provided in the options into both equation
A.\((4,3)\) \( x \ + \ y=0→4 \ + \ 3≠0 \)
B.\((5,4)\) \(x \ + \ y=0→5 \ + 4 ≠0\)
C.\((4,- \ 4)\) \(x \ + \ y=0→4 \ + \ ( -\ 4)=0\)
D.\((4,-\ 6)\) \(x \ + \ y=0→4 \ + \ (-\ 6)=0\)
Only option C is correct.
Method 2: Multiplying each side of \(x \ + \ y=0 \) by \(2\) gives \(2 \ x \ + \ 2 \ y=0\).
Then, adding the corresponding side of \( 2 \ x \ + \ 2 \ y=0\) and \(4 \ x \ - \ 2 \ y=24\) gives \( 6 \ x=24\) .
Dividing each side of \(6 \ x=24\) by \(6\) gives \(x=4\) .
Finally, substituting \(4\) for \(x\) in \( x \ + \ =0\) , or \( y= - \ 4 \).
Therefore, the solution to the given system of equations is \(( 4,-\ 4)\)

The correct answer is  \(28 \ x \ + \ 6\)
If \( f(x)=  3 \ x \ + 4\  (\ x \ +\ 1 ) \ +\ 2 \) , then find \(f(4 \ x) \)  by substituting   \(4 \ x\) for  every \(x\) in the function.
This gives:
\(f(4\ x) = 3 \ (4\ x \ ) \ + \ 4 \ (4 \ x \ + \ 1 \ ) \ + \ 2\),
It simplifies to: \(f(4 \ x \ )=3 \ (4 \ x \ ) \ + \ 4  \ (4 \ x \ + \ 1 ) \ + \ 2=12 \ x \ + \ 16 \ x \ + \ 4 \ + \ 2=28 \ x \ + \ 6\)

The correct answer is \((9,3)\)
First, find the equation of the line.
All lines through the origin are of the form \(y=m\)
so the equation is \(y=\frac {1}{3}\ x\) .
Of the given choices, only choice C \( (9 \ , \ 3)\) , satisfies this equation:
\( y=\frac{1}{3} \ x→\ 3 =\frac {1}{3} \ (9)=3\)

The correct answer is \(n^2 \ + \ 2 \ n \ + \ 10 \)
\((3 \ n ^2 \ + \ 2 \ n \ + \ 6 ) \ - \ (2 \ n^2 \ - \ 4  )\)
Add like terms together:
\(3 \ n^2 \ - \ 2 \ n^2=n^2\)
\(2 \ n \) doesn’t have like terms.
\(6 \ - \ ( - \ 4)=10\)
Combine these terms into one expression to find the answer:
\(n^2 \ + \ 2 \ n \ + \ 10 \)

The correct answer is \(23\) , \( 26\)
You can find the possible values of \(a\) and \(b\) in \((a \ x \ + \ 4) \ (b \ x \ + \ 3)\) by using the given equation \(a \ + \ b=7\)
and finding another equation that relates the variables \(a\) and \(b\).
Since \((a \ x \ + \ 4)\ (b \ x \ + \ 3)=10 \ x^2 \ + \ c \ x \ + \ 12\),
expand the left side of the equation to obtain
\(a \ b \ x^2 \ + \ 4 \ b \ x \ + \ 3 \ a \ x \ + \ 12=10 \ x^2 \ + \ c \ x \ + \ 12\)
Since \(a \ b\) is the coefficient of \(x^2\) on the left side of the equation and \(10\) is the coefficient of \(x^2\) on the right side of the equation,
it must be true that \(a \ b=10\)
The coefficient of \(x\) on the left side is \(4 \ b \ + \ 3 \ a\) and the coefficient of \(x\) in the right side is \(c\).
Then: \(4 \ b \ + \ 3 \ a=c\)
\(a \ + \ b=7\), then: \(a=7 \ - \ b\)
Now, plug in the value of a in the equation \(a \ b=10\).
Then:
\(a \ b=10→(7 \ - \ b) \ b=10→7 \ b \ - \ b^2=10\)
Add \( \ - \ 7 \ b \ + \ b^2\) both sides.
Then: \(b^2 \ - \ 7 \ b \ + \ 10=0\)
Solve for \(b\) using the factoring method.
\(b^2 \ - \ 7 \ b \ + \ 10=0→(b \ - \ 5)\ (b \ - \ 2)=0\)
Thus, either \(b=2\) and \(a = 5\), or \(b = 5\) and \(a = 2\).
If \(b = 2\) and \(a = 5\),
then
\(4 \ b \ + \ 3 \ a=c→4\ ( \ 2 \ ) \ + \ 3 \ ( \ 5\ )=c→c=23\)

The correct answer is \(\frac{( \ x \ - \ 5 \ ) \ ( \ x \ + \ 4 \ )}{( \ x \ - \ 5 \ ) \ + \ ( \ x \ + \ 4 \ )}\) 
To rewrite \(\frac{1}{{\frac{1}{x\ -\ 5}}\ +\ {\frac{1}{x\ +\ 4}}}\),
first simplify \({\frac{1}{x\ - \ 5}}\ + \ {\frac{1}{x\ +\ 4}}\).
\({\frac{1}{(x \ - \ 5)}+\frac{1}{( x \ + \ 4)}=\frac{1 \ (x \ + \ 4)}{(x \ - \ 5)(x \ + \ 4)}+\frac{1\ (x \ - \ 5 )}{(x \ + \ 4)\ (x \ - \ 5)}}=\frac {(x \ + \ 4) \ + \ (x \ - \ 5)}{(x \ + \ 4)\ (x \ - \ 5)}\)
Then
\(\frac{1}{{\frac{1}{x\ -\ 5}}\ +\ {\frac{1}{x\ +\ 4}}}\)=\(\frac {1}{\frac{( \ x \ + \ 4 \ ) \ + \ ( \ x \ - \ 5 \ ) }{( \ x \ + \ 4 \ )\ ( \ x \ - \ 5 \ )}}\)=\(\frac{( \ x \ - \ 5 \ ) \ ( \ x \ + \ 4 \ )}{( \ x \ - \ 5 \ ) \ + \ ( \ x \ + \ 4 \ )}\) . (Remember,\(\frac{1}{\frac{1}{x }}=x\))
This result is equivalent to the expression in choice A.

The correct answer is \(a \ > \ b \)
Since \((0 \ , \  0)\) is a solution to the system of inequalities, substituting \(0\) for \(x\) and \(0\) for \(y\)  in the given system must result in two true inequalities.
After this substitution, \(y \ < \ a \ - \ x\) becomes \(0 \ < \ a\) ,
and  \(y \ > \ x \ + \ b\) becomes  \(0 \  > \ b\).
Hence, \(a\)  is positive and \(b\) is negative.
Therefore, \(a \ > \ b\).

The correct answer is \( 6 \) , \( 6 \)
First find the slope of the line using the slope formula.
\(m=\frac {y_2 \ - \ y_1}{x_2 \ - \ x_1}\)
Substituting in the known information.
\((\ x_1 \ , \ y_1 )=(2 \ ,4 )\)
\(( \ x_2 \ ,\ y_2 \ )=( \ 4 \ , \ 5 \ )\)
\(m=\frac {5 \ - \ 4}{4 \ - \ 2}=\frac{1}{2}\)
Now the slope to find the equation of the line passing through these points.
\(y=m \ x \ + \ b\)
Choose one of the points and plug in the values of \(x\) and \(y\) in the equation to solve for \(b\).
Let’s choose point \((\ 4 \ , \ 5)\). Then:
\( \ y=m \ x \ + \ b→5=\frac{1}{2} \ (4) \ + \ b→5=2 \ + \ b→b=5 \ - \ 2=3\)
The equation of the line is: \(y=\frac{1}{2} \  x \ + \ 3\)
Now, plug in the points provided in the choices into the equation of the line.
\((\ 9\ ,\ 9\ )\) \(y=\frac {1}{2} \ + \ 3→9=\frac{1}{2}(\ 9 ) \ + \ 3→9=7.5\) This is NOT true.
\((\ 9\ ,\ 6\ )\) \(y=\frac{1}{2} x \ + \ 3→6=\frac{1}{2}\ (\ 9) \ + \ 3→6=7.5\) This is NOT true.
\((\ 6\ , \ 9\ )\) \(y=\frac{1}{2} x \ + \ 3→9=\frac{1}{2}\ ( 6) \ + \ 3→9=6\) This is NOT true.
\((\ 6\ , \ 6\ )\) \(y=\frac{1}{2} x \ + \ 3→6=\frac{1}{2}( \ 6) \ + \ 3→6=6\) This is true!
Therefore, the only point from the choices that lies on the line is \((6 , 6) \) .

The correct answer is \(10\)
The input value is \(5\). Then: \(x = 5\)
\(f(x) = x^2\ - \ 3 \ x → f(5) = 5^2 \ - \ 3 \ (5) = 25 \ - \ 15 = 10\)

The correct answer is  The \(y-\)intercept represents the starting height of \(6\) inches
To solve this problem, first recall the equation of \(a\) line: \(y=m \ x \ + \ b\)
Where \(m=\) slope
\(y=y-\)intercept
Remember that slope is the rate of change that occurs in a function and that the \(y-\)intercept is the \(y\) value corresponding to \(x=0\).
Since the height of John’s plant is \(6\) inches tall when he gets it. Time (or \(x\)) is zero.
The plant grows \(4\) inches per year.
Therefore, the rate of change of the plant’s height is \(4\).
The \(y-\)intercept represents the starting height of the plant which is \(6\) inches.

The correct answer is \(- \ 2\)
Multiplying each side of \(\frac{4}{x}=\frac{12}{x \ - \ 8}\) by \(x \ (x \ - \ 8)\) gives \(4 \ ( \ x \ - \ 8 \ )=12 \ ( \ x \ )\),
distributing the \( 4 \) over the values within the parentheses yields \( x \ - \ 8=3 \ x \) or \(x=- \ 4\).
Therefore, the value of  \(\frac{x}{2}\)=\(\frac{(- \ 4 )}{2}=- \ 2\).

The correct answer is \(7\)
Adding \(6\) to each side of the inequality \(4 \ n \ - \ 3 \ ≥ \ 1\) yields the inequality \(4 \ n \ + \ 3 \ ≥ \ 7\).
Therefore, the least possible value of \(4 \ n \ + \ 3\) is \(7\).

The correct answer is \(-\frac{8}{7}\)
Since\(f(x)\) is linear function with a negative slop, then when \(x=-\ 2 \) , \(f(x)\) is maximum and when \(x=3\) , \(f(x)\) is minimum.
Then the ratio of the minimum value to the maximum value of the function is:\(\frac{f(3)}{f(-2)}\)=\(\frac{- \ 3 \ ( \ 3 \ ) \ + \ 1 \ }{- \ 3 \ ( \ - \ 2) \ + \ 1}\)=\(\frac  {- \ 8}{7}\)=\(-\frac {8}{7}\)

The correct answer is \(2\)
 Method 1: There can be \(0\), \(1\), or \(2\) solutions to a quadratic equation.
In standard form, a quadratic equation is written as: \(a \ x^2 \ + \ b \ x \ + \ c=0\)
For the quadratic equation, the expression \(b^2 \ - \ 4 \ a \ c\) is called discriminant.
If discriminant is positive, there are \(2\) distinct solutions for the quadratic equation.
If discriminant is \(0\), there is one solution for the quadratic equation and if it is negative the equation does not have any solutions.
To find number of solutions for \(x^2=4 \ x \ - \ 3\), first, rewrite it \(a \ s \ x^2 \ - \ 4 \ x \ + \ 3=0\).
Find the value of the discriminant. \(b^2\ -\ 4 \ a \ c=( - \ 4)^2 \ - \ 4 \ (1) \ (3)=16 \ - \ 12=4\)
Since the discriminant is positive, the quadratic equation has two distinct solutions.

The correct answer is  \(\frac{11}{30}\)
Of the \(30\) employees, there are \(5\) females under age \(45\) and \(6\) males age \(45\) or older.
Therefore, the probability that the person selected will be either a female under age \(45\) or a male age \(45\) or older is:\(\frac{5}{30} \ + \frac {6}{30}=\frac{11}{30}\)

The correct answer is \(86\)
In the figure angle A is labeled \((3 \ x \ - \ 2)\) and it measures \(37\).
Thus, \(3 \ x \ - \ 2=37\) and \(3 \ x=39\) or \(x=13\).
That means that angle B, which is labeled \((5 \ x)\), must measure \(5 \ × \ 13=65\).
Since the three angles of a triangle must add up to \(180\) , \(37 \ + \ 65 \ + \ y \ - \ 8=180\) , then:
\(y \ + \ 94=108→y=180 \ - \ 94=86\)

The correct answer is \(22\)
Substituting \(6\) for \(x\) and \(14\) for \(y\) in \(y = n \ + \ 2\) gives \(14=( \ n \ )\ ( \ 6 \ )+2 \) ,
which gives \(n=2\). Hence, \(y=2 \ x \ + \ 2\).
Therefore, when\( = 10\) , the value of \(y\) is
\(y=( \ 2 \ ) \ ( \ 10 \ ) \ + \ 2 = 22\).

The correct answer is \(-\ 2\)
Subtracting \(2 \ x \) and adding \(5\) to both sides of \(2 \ x \  - \  5 \ ≥ \ 3 \ x \ - \ 1\) gives \(- \ 4 \ ≥ \ x\).
Therefore, \(x\) is a solution to \(2 \ x \ - \ 5 \ ≥ \ 3 \ x \ - \ 1\) if and only if \(x\) is less than or equal to \(- \ 4\) and
\(x\) is NOT a solution to \(2 \ x \ - \ 5 \ ≥ \ 3 \ x  \ - 1\) if and only if \(x\) is greater than \(- \ 4\).
Of the choices given, only \(- \ 2\) is greater than \(- \ 4\) and,
therefore, cannot be a value of \(x\).

The correct answer is \(12\)
Given the two equations,
substitute the numerical value of \(a\) into the second equation to solve for \( x\ . \ a=\sqrt{3}\),
\(4\ a=\sqrt{4 \ x}\)
Substituting the numerical value for ainto the equation with \(x\) is as follows.
\(4 \ (\sqrt{3})=\sqrt{4 \ x}\),
From here, distribute the \(4\) . \(4\sqrt{3}=\sqrt { \ 4 \ x}\)
Now square both side of the equation.
\((4\sqrt{3})^2=(\sqrt{4 \ x})^2\)
Remember to square both terms within the parentheses.
Also, recall that squaring a square root sign cancels them out.
\(4^2 \sqrt{3}^2=4 \ x\) ,
\(16 \ (3)=4 \ x\) ,
\(48=4 \ x\) ,
\(x=12\)

The correct answer is \(1 \ , \ 3\)
First square both sides of the equation to get \(4 \ m-3=m^2\)
Subtracting both sides by \(4 \ m \ - \ 3\) gives us the equation \( \ m^2 \ - \ 4 \ m \ + \ 3=0\)
Here you can solve the quadratic equation by factoring to get \((\ m \ - \ 1) \ ( \ m \ - \ 3  )=0\)
For the expression \(( \ m \ - \ 1) \ ( \ m \ - \ 3  )\) to equal zero, \(m=1\) or \(m=3\)

The correct answer is \(\frac{5}{4}\)
To solve the equation for \(y\) , multiply both sides of the equation by the reciprocal of  \(\frac{6}{5}\) , which is  \(\frac{5}{6}\) ,
 this gives \(\frac{(5)}{(6)}×\frac {6}{5} y= \frac {3}{2} \ × \frac {(5)}{(6)}\), which simplifies to \(y=\frac{15}{12}=\frac{5}{4}\).

The correct answer is \(720\) m
Because Jack walks \(30\) meters in \(15\) seconds, and \(6\) minutes is equal to \(360\) seconds,
use the proportion to solve.
\(\frac{30\  meters}{15\ s ec}=\frac{x \ meters}{360 \ sec}\)
The proportion can be simplified to \(\frac{30}{15}=\frac{x}{360}\) then each side of the equation can be multiplied by \(360\) ,
giving \(\frac{(360)(30)}{15}=x=720\).
Therefore, \(720\) meters is the distance Jack will walk in \(6\) minutes.

A zero of a function corresponds to an \(x-\)intercept of the graph of the function in the \(x \ y-\)plane.
Therefore, the graph of the function \(g ()\), which has three distinct zeros, must have three \(x-\)intercepts.
Only the graph in choice C has three \(x-\)intercepts.

The correct answer is \(3\)
The fastest way to find the answer is to pick numbers.
Pick a number for that has a remainder of \(2\) when divided by \(8\), such as \(10\). 
Increase the number you picked by \(9\). In this case \(10 \ + \ 9 \ =19\) .
Now divide \(19\) by \(8\), which gives you remainder \(3\). Therefore, the answer is \( 3\).

The answer is  \(c=0.35 \ ( \ 60 \ h)\)
\($0.35\) per minute to use car.
This per-minute rate can be converted to the hourly rate using the conversion \(1\) hour = \(60\) minutes, as shown below.
\(\frac{0.35}{minute} \ × \frac {60 minutes}{1 hours}=\frac{$ \ ( \ 0.35 \ × \ 60 \ )}{ hour}\)
Thus, the car costs \($( \ 0.35 \ × \ 60 \ )\) per hour.
Therefore, the cost \(c\) , in dollars, for h hours of use is \(c=( \ 0.35 \ × \ 60) \ h \),
Which is equivalent to \(c=0.35 \ (60 \ h)\)

The correct answer is \(100\)
The best way to deal with changing averages is to use the sum.
Use the old average to figure out the total of the first \(4\) scores:
Sum of first \(4\) scores: \((4) \ (90) = 360\)
Use the new average to figure out the total she needs after the \(5^{th}\) score:
Sum of score: \((5) \ (92) = 460\) 
To get her sum from \(360\) to \(460\) ,
Mary needs to score \(460 \ - \ 360=100\).

The correct answer is \(2\)
To solve a quadratic equation, put it in the \({a \ x}^2 \ + \ b \ x \ + \ c=0 \) form, factor the left side,
and set each factor equal to \(0\) separately to get the two solutions.
To solve \({x}^2=5 \ x \ - \ 4\) , first, rewrite it as \({x}^2 \ - \ 5 \ x \ + \ 4=0\).
Then factor the left side: \({x}^2 \ - \ 5 \ x \ + \ 4=0\) , \((x \ - \ 4) \ (x \ - \ 1)=0\)
\(x=1\) Or \(x=4\) , There are two solutions for the equation.

The correct answer is \(31,752\)
\(30\%\) of the books are Mathematics books and \(15\%\) of the books are English books. Thus, number of Mathematics books: \(0.3 \ × \ 840=252\)
Number of English books: \(0.15 \ × \ 840=126\)
The product of number of Mathematics and number of English books: \(252 \ × \ 126=31,752\)

The correct answer is \(108^\circ\), \(54^\circ\)
All central angles in a circle sum up to \(360\) degrees.
Thus, the angle \(α\)  is: \(0.3 \ × \ 360=108^\circ\)

The correct answer is \(120\)
According to the graph, \(50\%\) of the books are in the Mathematics and Chemistry sections.
Therefore, there are \(420\) books in these two sections.
\(0.50 \ × \ 840 = 420\)
\(γ \ + \ α=420\), and \(γ=\frac{2}{5} \ α\)
Replace\( γ\) by \(\frac{2}{5} \ α\) in the first equation.
\(γ \ + \ α=420 → \frac{2}{5} α \ + \ α=420 → \frac{7}{5} α=420\) → multiply both sides by \(\frac{5}{7}\)
\(\frac{(5)}{(7)} \frac{7}{5} \ α = 420 \ ×  \frac{(5)}{(7)} → α = \frac{420 \ × \ 5}{7}=300\)
\(α=300 → γ =  \frac{2}{5}\ α  → γ = \frac {2}{5} \ × \ 300=120\)
There are \(120\) books in the Chemistry section.

The correct answer is \(6 \sqrt{2}\)
The line passes through the origin, \((6 \ , \ m)\) and \((m \ , \ 12)\).
Any two of these points can be used to find the slope of the line.
Since the line passes through \((0 \ , \ 0)\) and \((6 \ , \ m)\),
the slope of the line is equal to\(\frac{m \ - \ 0}{6 \ - \ 0}\)=\(\frac{m}{6}\).
Similarly, since the line passes through \((0 \ , \ 0)\) and \((m \ , \ 12)\) ,
 the slope of the line is equal to  \(\frac{12 \ - \ 0}{m \ - \ 0}\)=\(\frac{12}{m}\).
since each expression gives the slope of the same line, it must be true that \(\frac{m}{6}=\frac{12}{m}\)
Using cross multiplication gives
\(\frac{m}{6}=\frac{12}{m}→m^2=72→m=± \sqrt{72}=± \sqrt{36 \ × \ 2}=± \sqrt{36} \ × \sqrt{2}=± \ 6 \sqrt{2}\)

The correct answer is \(6\)
It is given that \(g(5)=4\). Therefore, to find the value of \(f(g(5))\), then \(f(g(5))=f(4)=6\)

The correct answer is  \(16\sqrt{3 }\) cm\(^2\)
Area of the triangle is: \(\frac{1}{2}\) AD\(×\)BC and AD is perpendicular to BC.
Triangle ADC is a \( 30^° \ - \ 60^° \ - \ 90^°\) right triangle.
The relationship among all sides of right triangle \(30^° \ - \ 60^° \ - \ 90^°\) is provided in the following triangle:
In this triangle, the opposite side of \(30^°\) angle is half of the hypotenuse.
And the opposite side of \(60^°\) is opposite of  \(30^° \ × \sqrt{3}\)
CD = \(4\) , then AD = \(4 \ ×\sqrt{3}\)
Area of the triangle ABC is :\(\frac {1}{2}\) AD×BC =\(\frac {1}{2}  4\sqrt {3} \ × \ 8=16\sqrt{3}\)

The correct answer is \(32\)
It is given that \(g(5)=8\).
Therefore, to find the value of \(f(g(5) )\), substitute \(8\) for \(g(5)\).
\(f(g(5) )=f(8)=32\).

The correct answer is  \(c=- \ 2\ ,\ d=6\)
Substituting \(5\) for yin \(y=c \ x^2 \ + \ d\) gives \(5=c \ x^2 \ + \ d \) which can be rewritten as \(5 \ - \ d=c \ x^2\) .
Since \(y = 5\) is one of the equations in the given system, any solution \(x\) of \(5 \ - \ d=c \ x^2\) corresponds to the solution \((x \ , \ 5)\) of the given system.
Since the square of a real number is always nonnegative, and a positive number has two square roots,
the equation \(5 \ - \ d=c \ x^2\) will have two solutions for \(x\) if and only if \((1) \ c \ > \ 0\) and \(d \ < \ 5\) or \((2) \ c \ < \ 0\) and \(d \ > \ 5\).
Of the values for cand dgiven in the choices, only \(c=- \ 2\), \(d=6\) satisfy one of these pairs of conditions.
Alternatively, if \(c=- \ 2\) and \(d=6\) , then the second equation would be
\(y=- \ 2 \ x^2 \ + \ 6\)
The equation above has two real answer.

The correct answer is  \(y \ + \ 4 \ x=z\)
\(x\) and \(z\) are colinear. \(y\) and \(5 \ x\) are colinear. Therefore,
\(x \ + \ z=y \ + \ 5 \ x\) ,subtractxfrombothsides,then,  \(z=y \ + \ 4 \ x\)

The correct answer is \(29\)
Here we can substitute \(8\) for \(x\) in the equation.
Thus, \(y \ - \ 3=2 \ (8 \ + \ 5)\) , \(- \ 3=26\)
Adding \(3\) to both side of the equation:
\(y=26 \ + \ 3\) , \(y=29\)

The correct answer is I and III only
Let’s review the options:
I. \(|a| \ < \ 1→- \ 1 \ < \ a \ < \ 1\)
Multiply all sides by \(b\). Since, \(b \ > \ 0→- \ b \ < \ b \ a \ < \ b\)
II. Since , \(- \ 1 \ < \ a \ < \ 1\) , and \(a \ < \ 0→- \ a \ > \ a^2 \ > \ a\)  (plug in \(\frac{- \ 1}{2}\), and check!)
III. \(- \ 1 \ < \ a \ < \ 1\) ,multiply ll sides by \(2\) , then:
\(- \ 2 \ < \ 2 \ a \ < \ 2\) , subtract \(3\) from all sides,then:
\(- \ 2 \ - \ 3 \ < \ 2 a \ - \ 3 \ < \ 2 \ - \ 3→- \ 5 \ < \ 2 \ a \ - \ 3 \ < \ - \ 1\)
I and III are correct.

The correct answer is  \(a \ > \ 1\)
The equation can be rewritten as
\(c \ - \ d=a \ c\) →(divide both sides by \(c\) ) \( 1 \ - \frac {d}{c}=a\) ,
since \(c \ < 0 \) and \(d \ > \ 0\) , the value of \(- \frac{d}{c}\) is positive.
Therefore, \(1\) plus a positive number is positive. \(a\)  must be greater than \(1\).
\(a \ > \ 1\)

The correct answer is \(8\)
Squaring both sides of the equation gives \(2 \ m \ + \ 48=m^2\)
Subtracting both sides by \(2 \ m \ + \ 48\) gives us the equation \(m^2 \ - \ 2 \ m \ - \ 48=0\)
Here you can solve the quadratic by factoring to get \((m \ - \ 8) \ (m \ + \ 6)=0\)
For the expression \((m \ - \ 8) \ (m \ + \ 6)\) to equal zero, \(m=8\)  or  \(m=- \ 6\)
Since \(m\) is a positive integer, \(8\) is the answer.

The correct answer is \(3\)
Since we are dealing with an absolute value, \(f(a)=20\) means that either \(11 \ - \ a^2=20\)  or
\(11 \ - \ a^2=- \ 20\)
Let’s start with the positive value \((20)\) and see what we get.
 If \(11 \ - \ a^2=20\) , then \(a^2=9\)
Taking the square root, we get \(a=3\) or \(- \ 3\)
On the other hand, if \(11 \ - \ a^2=- \ 20\),
 then \(a=\sqrt {- \ 31}\)
Notice that the question states that \(a\) is a positive integer, therefore the answer is \(3\).

The correct answer is \(20\%\) 
Use this formula: Percent of Change
\(\frac{New Value \ - \ Old Value}{Old Value}×100\%\)
\(\frac{16000 \ - \ 20000}{20000} \ × \ 100\%=20\%\) and \(\frac{12800 \ - \ 16000}{16000} \ × \ 100 \%=20\%\)

The correct answer is \(729\) cm\(^3\)
If the length of the box is \(27\) , then the width of the box is one third of it, \(9\) , and the height of the box is \(3\) (one third of the width). The volume of the box is:
\(V = lwh = (27) \ (9) \ (3) = 729\)

The correct answer is \(12\)
Let \(x\) represent the number of liters of the \(30\%\) solution.
The amount of salt in the \(30\%\) solution \((0.30\ x)\) plus the amount of salt in the \(75\%\) solution \((0.75) \ × ( \ 5 )\) must be equal to the amount of salt in the \(39\%\) mixture \((0.39 \ × \ (x \ + \ 5))\).
Write the equation and solve for \(x\).
\(0.30 \ x \ + \ 0.75 \ ( 3) = 0.39 \ (x \ + \ 3) →0.30 \ x \ + \ 2.25 = 0.39 \ x \ + \ 1.17 →
 0.39 \ x \ - \ 0.30 \ x = 2.25 \ - \ 1.17→0.09 \ x = 1.08 → x = \frac{1.08}{0.09}=12\)

The correct answer is \(50\) melis
Use the information provided in the question to draw the shape.
Use Pythagorean Theorem: \(a^2 \ + \ b^2=c^2\)
\(40^2 \ + \ 30^2\)= \(c^2\)⇒ \(1600 \ + \ 900\) = \(c^2\) ⇒ \(2500\) = \(c^2\) ⇒ \(c=50\)

The correct answer is \(360\)
One of the four numbers is \(x\) ;
let the other three numbers be \(y\) , \(z\) and \(w\).
Since the sum of four numbers is \(600\) , the equation \(x \ + \ y \ + \ z \ + \ w = 600\) is true.
The statement that \(x\) is \(50\%\) more than the sum of the other three numbers can be represented as
\(x = 1.5 \ (y \ + \ w)\) or \(\frac{x}{1.5} =  y \ + \ z \ + \ w → \frac{2x}{3} = y \ + \ z \ + \ w\)
Substituting the value \(y \ + \ z \ + \ w\) in the equation \(x \ + \ y \ + \ z \ + \ w=600\)
gives \(x \ + \frac {2 \ x}{3} = 600 →  \frac{5 \ x}{3} =  600  →  5 \ x = 1,800 →  x  =  \frac{1,800}{5} = 360\)

The correct answer is \(570\)
This is a simple matter of substituting values for variables.
We are given that the \(50\) cars were washed today, therefore we can substitute that for \(a\).
Giving us the expression \(\frac{40\ (50)\ -\ 500}{50} \ + \ b\)
We are also given that the profit was \($600\), which we can substitute for \(f(a)\).
Which gives us the equation \(600= \frac{40\ (50)\ -\ 500}{50} \ + \ b\)
Simplifying the fraction gives us the equation \(600=30 \ + \ b\)
And subtracting both sides of the equation by \(30\) gives us \(b=570\), which is the answer.

The correct answer is \(7.32\)
The weight of \(12.2\) meters of this rope is: \(12.2 \ × \ 600 \) g = \(7,320\) g
\(1\) kg \(= 1,000 \) g , therefore , \(7,320 \) g \(÷ \ 1000 = 7.32 \) kg

The correct answer is \(40\)
The area of ∆BED is \(16\), then: \(\frac{4 × \ AB}{2}\)=\(16→4 \ × \ AB=32→AB=8\)
The area of ∆BDF is \(18\), then: \(\frac{3×BC } {2}=18→3 \ × \ BC=36→BC=12\)
The perimeter of the rectangle is = \(2 \ × \ (8 \ + \ 12)=40\)