Szczegółowe wyjaśnienie:
[tex]a)\ |x|\geq7\Rightarrow x\geq7\ \vee\ x\leq-7\\\\\huge\boxed{x\in\left(-\infty,\ -7\right>\ \cup\ \left<7,\ \infty\right)}[/tex]
[tex]b)\ |2x|>\dfrac{8}{9}\Rightarrow2x>\dfrac{8}{9}\ \vee\ 2x<-\dfrac{8}{9}\qquad|:2\\\\x>\dfrac{4}{9}\ \vee\ x<-\dfrac{4}{9}\\\\\huge\boxed{x\in\left(-\infty,\ -\dfrac{4}{9}\right)\ \cup\ \left(\dfrac{4}{9},\ \infty\right)}[/tex]
[tex]c)\ \left|x+\dfrac{1}{3}\right|\geq3\Rightarrow x+\dfrac{1}{3}\geq3\ \vee\ x+\dfrac{1}{3}\leq-3\qquad|-\dfrac{1}{3}\\\\x\geq2\dfrac{2}{3}\ \vee\ x\leq-3\dfrac{1}{3}\\\\\huge\boxed{x\in\left(-\infty,\ -3\dfrac{1}{3}\right>\ \cup\ \left<2\dfrac{2}{3},\ \infty\right)}[/tex]
[tex]d)\ \left|\dfrac{2}{5}x-6\right|>1\Rightarrow\dfrac{2}{5}x-6>1\ \vee\ \dfrac{2}{5}x-6<-1\qquad|+6\\\\\dfrac{2}{5}x>7\ \vee\ \dfrac{2}{5}x<5\qquad|\cdot\dfrac{5}{2}\\\\x>\dfrac{35}{2}\ \vee\ x<\dfrac{25}{2}\\\\x>17,5\ \vee\ x<12,5\\\\\huge\boxed{x\in(-\infty;\ 12,5)\ \cup\ (17,5;\ \infty)}[/tex]
a) |x| ≥ 7
x ≥ 7, x ≥ 0
-x ≥ 7, x < 0
x∈ [7, + ∞>
x ≤ -7, x < 0
x∈ [7, + ∞>
x∈ < -∞, -7]
x∈ < -∞, -7] ∪ [7, + ∞>
b) |2x| > [tex]\frac{8}{9}[/tex]
2 × |x| > [tex]\frac{8}{9}[/tex]
2x > [tex]\frac{8}{9}[/tex], x ≥ 0
2 × (-x) > [tex]\frac{8}{9}[/tex], x < 0
x > [tex]\frac{4}{9}[/tex], x ≥ 0
x < - [tex]\frac{4}{9}[/tex], x < 0
x∈ < [tex]\frac{4}{9}[/tex], + ∞ >
x∈ < - ∞, - [tex]\frac{4}{9}[/tex] >
x∈ < - ∞, - [tex]\frac{4}{9}[/tex] > ∪ < [tex]\frac{4}{9}[/tex], + ∞ >
c) |x + [tex]\frac{1}{3}[/tex]| ≥ 3
x + [tex]\frac{1}{3}[/tex] ≥ 3, x + [tex]\frac{1}{3}[/tex] ≥ 0
- (x + [tex]\frac{1}{3}[/tex]) ≥ 3, x + [tex]\frac{1}{3}[/tex] < 0
x ≥ [tex]\frac{8}{3}[/tex], x ≥ - [tex]\frac{1}{3}[/tex]
x ≤ - [tex]\frac{10}{3}[/tex], x < - [tex]\frac{1}{3}[/tex]
x∈ [ [tex]\frac{8}{3}[/tex], + ∞ >
x∈ < - ∞, - [tex]\frac{10}{3}[/tex] ]
x∈ < - ∞, - [tex]\frac{10}{3}[/tex] ] ∪ [ [tex]\frac{8}{3}[/tex], + ∞ >
d) | [tex]\frac{2}{5}[/tex] x - 6| > 1
[tex]\frac{2}{5}[/tex] x - 6 > 1, [tex]\frac{2}{5}[/tex] x - 6 ≥ 0
- ( [tex]\frac{2}{5}[/tex] x - 6 ) > 1, [tex]\frac{2}{5}[/tex] x - 6 < 0
x > [tex]\frac{35}{2}[/tex], x ≥ 15
x < [tex]\frac{25}{2}[/tex], x < 15
x∈ < [tex]\frac{35}{2}[/tex], + ∞ >
x∈ < - ∞, [tex]\frac{25}{2}[/tex] >
x∈ < - ∞, [tex]\frac{25}{2}[/tex] > ∪ < [tex]\frac{35}{2}[/tex], + ∞ >
Mam nadzieję, że pomogłam. Liczę na naj :)