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Matemáticas 3 · Tema 13

Desigualdades

Video

Ejemplos

Ejemplo 1
−15x+20<−6x+2-15x + 20 < -6x + 2
−15x+6x<+2−20−9x<−18x>−18−9\begin{aligned} -15x + 6x &< +2 - 20 \\ -9x &< -18 \\ x &> \dfrac{-18}{-9} \end{aligned}
x>2x > 2

intervalo

(2,∞)(2, \infty)
-3-2-1123
Ejemplo 2
20x−5≤10x−2520x - 5 \leq 10x - 25
20x−10x≤−25+510x≤−20x≤−2010\begin{aligned} 20x - 10x &\leq -25 + 5 \\ 10x &\leq -20 \\ x &\leq \dfrac{-20}{10} \end{aligned}
x≤−2x \leq -2

intervalo

(−∞,−2](-\infty, -2]
-3-2-1123

Cuando se divide entre un número negativo, la desigualdad se invierte.

Otros tipos de intervalos

x≥2x \geq 2
[2,∞)[2, \infty)
-1123
x<3x < 3
(−∞,3)(-\infty, 3)
-1123
1<x≤41 < x \leq 4
(1,4](1, 4]
1234

Ejercicios

Bloque 1
4x−3≥2x−94x - 3 \geq 2x - 9
3x−4>7x+83x - 4 > 7x + 8
Bloque 2
−2x+6≤−5x−9-2x + 6 \leq -5x - 9
5x+2<−x−105x + 2 < -x - 10
Bloque 3
x−6>−x+4x - 6 > -x + 4
−5x−3≤2x+11-5x - 3 \leq 2x + 11

Respuestas

▼

Bloque 1

4x−3≥2x−94x - 3 \geq 2x - 9
4x−2x≥−9+32x≥−6x≥−62\begin{aligned} 4x - 2x &\geq -9 + 3 \\ 2x &\geq -6 \\ x &\geq \dfrac{-6}{2} \end{aligned}
x≥−3x \geq -3

intervalo

[−3,∞)[-3, \infty)
-3-2-1123
3x−4>7x+83x - 4 > 7x + 8
3x−7x>8+4−4x>12x<12−4\begin{aligned} 3x - 7x &> 8 + 4 \\ -4x &> 12 \\ x &< \dfrac{12}{-4} \end{aligned}
x<−3x < -3

intervalo

(−∞,−3)(-\infty, -3)
-4-3-2-11

Bloque 2

−2x+6≤−5x−9-2x + 6 \leq -5x - 9
−2x+5x≤−9−63x≤−15x≤−153\begin{aligned} -2x + 5x &\leq -9 - 6 \\ 3x &\leq -15 \\ x &\leq \dfrac{-15}{3} \end{aligned}
x≤−5x \leq -5

intervalo

(−∞,−5](-\infty, -5]
-6-5-4-3-2-1
5x+2<−x−105x + 2 < -x - 10
5x+x<−10−26x<−12x<−126\begin{aligned} 5x + x &< -10 - 2 \\ 6x &< -12 \\ x &< \dfrac{-12}{6} \end{aligned}
x<−2x < -2

intervalo

(−∞,−2)(-\infty, -2)
-3-2-11

Bloque 3

x−6>−x+4x - 6 > -x + 4
x+x>4+62x>10x>102\begin{aligned} x + x &> 4 + 6 \\ 2x &> 10 \\ x &> \dfrac{10}{2} \end{aligned}
x>5x > 5

intervalo

(5,∞)(5, \infty)
123456
−5x−3≤2x+11-5x - 3 \leq 2x + 11
−5x−2x≤11+3−7x≤14x≥14−7\begin{aligned} -5x - 2x &\leq 11 + 3 \\ -7x &\leq 14 \\ x &\geq \dfrac{14}{-7} \end{aligned}
x≥−2x \geq -2

intervalo

[−2,∞)[-2, \infty)
-2-112

Ejercicios extra

▼
Bloque 1
3x+5>x−13x + 5 > x - 1
−4x+7≤2x−5-4x + 7 \leq 2x - 5
5x−8<2x+45x - 8 < 2x + 4
−2x−9≥3x+6-2x - 9 \geq 3x + 6
6x+1≤4x−76x + 1 \leq 4x - 7
Bloque 2
−x+10>4x−5-x + 10 > 4x - 5
7x−3≥5x+97x - 3 \geq 5x + 9
−3x+2<−6x−10-3x + 2 < -6x - 10
8x+5>3x−208x + 5 > 3x - 20
−5x−4≤−2x+8-5x - 4 \leq -2x + 8
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