Equations: math exercises in 2nd grade corrected in PDF.

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1 Exercise series equations

An equation is defined as a relationship containing one or more variables. Thus, solving the equation consists in determining the values that the variable can take so that the equality can be true. On the other hand, the variable is also called unknown. With more practice on the equations, you will master this chapter completely.

Exercise series equations

Exercise 1:

$(E_1)\,:\,(0,1\,x\,-\,1)(0,2\,x\,-\,2)(0,3\,x-\,3)(0,04\,x-0,4)\,=\,0$

(E2): $\frac{2x+3}{5x-1}$ = 2

(E3) : 4 x – 0.8 = 2 – 1.6 x

(E4): $\frac{3}{x}$ = $\frac{x}{5}$

(E5) : (x – 2⁾² = $\frac{1}{16}$ (5 – 2 x⁾²

(E6) : $\frac{x-4}{x-2}=\frac{x+2}{x}$

(E7) : (x + 1)(3 – 2 x) = 4 ^x2 – 9

(E8) : $\frac{x^2}{1-2x}$ = -1

(E9) : (x + 2⁾² = 2(^x2 – 4)

(E10) : $\frac{x^2+x+1}{2x-3}\,=\frac{1}{2}$

Exercise 2:

Solve in $\mathbb{R}$ the following equations:

a) $\frac{1-x\,}{4}-\frac{3x-2\,}{2}=\frac{2x+5\,}{6}$

(We will show that this equation is equivalent to: )

c) $3x-1=\frac{4}{3x-1}$

(It will be shown that this equation is equivalent to: $\frac{(3x-3)(3x+1)\,}{3x-1}=0$ .

Exercise 3:

Factor using a remarkable identity.

Exercise 4:

Here is the representative curve of a function f defined on [0;7].

Estimate the solutions of the following equations.

a) f(x)= 2 b) f(x) = 0 c) f(x) = – 1 d) f(x) = 1

Exercise 5:

For each of the functions whose expressions are given below,

try to establish the largest possible set of definitions.

$a)f(x)=\frac{5+x}{10-x}$ $b)g(x)=2\sqrt{x}+3$

$c)h(x)=\frac{3x+x^2}{2}$ $d)i(x)=4x+\frac{1}{x}$

Exercise 6:

Consider the representative curves of the inverse function, denoted f, and

of the affine function g defined x on R by g(x) = 2x + 1.

They are plotted in the marker below.

1. Locate the curves associated with the two functions.

2. Solve graphically the equation $\frac{1}{x}=\,2x\,+1$ .

3. a) Expand the expression (2x – 1)(x+ 1).
b) Find algebraically the results obtained in question 2.

Exercise 7:

Are the following equivalences true or false? (We will justify, and if the equivalence is false, we will add to the equation on the right what is necessary to make it equivalent to the equation on the left)

1) $(x\,+\,5)(x\,+\,1)\,=\,(3\,x-\,2)(x\,+\,1)$ $\Leftrightarrow$ $x\,+\,5\,=\,3\,x-\,2$

2) $(x\,+\,3)(x^2\,+\,1)\,=\,(x^2\,+\,1)(4\,x-1)$ $\Leftrightarrow$ $x\,+\,3\,=\,4\,x-1$

3) $(2\,x-3)^2\,=\,(3\,x-1)^2$ $\Leftrightarrow$ $2\,x-3\,=\,3\,x-1$

Exercise 8:

Solve in $\mathbb{R}$ the following equations.

$b)\,(2x+3)(4x-5)=0$

Exercise 9:

Solve in $\mathbb{R}$ the following equations.

Exercise 10:

Solve in $\mathbb{R}$ the following equations.

Exercise 11:

Solve in $\mathbb{R}$ the following equations.

Exercise 12:

Solve in $\mathbb{R}$ the following equations.

$a)\sqrt{x}=12$ $b)\,\sqrt{x}=-2$

$c)\,\sqrt{x}=11,5$ $d)3\sqrt{x}=21$

Exercise 13:

Solve in $\mathbb{R}$ the following equations.

$a)\frac{2x-1}{x+6}=1$ $b)\frac{4}{2x+6}=9$

$c)\frac{2x}{x-4}=-3$ $d)\frac{x+1}{x-1}=\frac{1}{2}$

Exercise 14:

The evolution of a population of bacteria is studied in a certain environment.
The number of bacteria in thousands was modeled as a function of time
elapsed in days over the first ten days of study by the function N defined
by $N(t)=(0,5t+\,1)^2$ for any real number $t\,\in\,%5B0;10%5D$ .