Question

Контрольная работа по теме уравнения и неравенства

Answer 1

1. 1) $x = 0,5$ ; 2) $x = 4$ ; 3) $x = -10$ ; 4) $x$₁ $=0$ ; $x$₂ $= log$₇$(5)$ ; 5) $x = -\frac{3}{7}$ ; 6) $x = 1$

2. 1) $x$ ∈ $[-1, +$ ∞ $)$ ; 2) $x$ ∈ $(\frac{9}{5} , 9]$

Пошаговое объяснение:

1.

1)

$\sqrt{x^2+4x} = 2-x$

$x^2+4x=4-4x+x^2$

$4x=4-4x$

$4x+4x=4$

$8x=4$

$x = \frac{1}{2}$

$\sqrt{(\frac{1}{2})^{2}+4*\frac{1}{2}} = 2 - \frac{1}{2}$

$\frac{3}{2} = \frac{3}{2}$

$x = \frac{1}{2}$ или $0,5$

2)

$2^x - 2^{x-4} = 15$

$(2^4-1) * 2^{x-4} = 15$

$(16-1) * 2^{x-4} = 15$

$15*2^{x-4} = 15$

$2^{x-4} = 1$

$2^{x-4}=2^0$

$x - 4 = 0$

$x = 4$

3)

$(\sqrt{3})^{x+4}= \frac{1}{27}$

$3^{\frac{1}{2}*(x+4)} = 3^{-3}$

$3^{\frac{1}{2}x+2} = 3^{-3}$

$\frac{1}{2} x + 2 = -3$

$x+4=-6$

$x = -6-4$

$x = -10$

4)

$7^{2x}-6*7^x+5=0$

$(7^x)^2-6*7^x+5=0$

$t^2-6t+5=0$

$t = 1$

$t = 5$

$7^x=1$

$7^x = 5$

$x = 0$

$x = log$₇$(5)$

$x$₁ $= 0$ ; $x$₂ $= log$₇$(5)$

5)

$log$₀,₅$(1-7x)=-2$

$log$₀,₅$(1-7x) = -2, x < \frac{1}{7}$

$1-7x=0,5^{-2}$

$1-7x=(\frac{1}{2})^{-2}$

$1-7x=2^2$

$1-7x=4$

$-7x=4-1$

$-7x=3$

$x = -\frac{3}{7}, x$

$x = -\frac{3}{7}$

6)

$log$₂$(x+1)+log$₂$(x+3) = 3$

$log$₂$(x+1)+log$₂$(x+3)=3, x$ ∈ $(-1,+$ ∞$)$

$log$₂$((x+1)(x+3))=3$

$log$₂$(x^2+3x+x+3)=3$

$x^2+3x+x+3=2^3$

$x^2+4x+3=8$

$x^2+4x+3-8=0$

$x^2+5x-x-5=0$

$x(x+5)-(x+5)=0$

$(x+5)(x-1)=0$

$x+5=0$

$x - 1 = 0$

$x = -5, x$ ∈ $(-1, +$ ∞$)$

$x = 1, x$ ∈ $(-1, +$ ∞$)$

$x = 1$

2.

1)

$0,5^{4x-3}\geq 0,5^{6x-1}$

$4x-3\leq 6x-1$

$4x-6x \leq -1+3$

$-2x\leq 2$

$x \geq -1$

$x$ ∈ $[-1, +$ ∞$)$

2)

см. фото.