Question

Решите неравенства a) X^2-49>0 б)x^2+6X<0 в)2X^2-x-6≤0 Г)X^2-6x+9≤0 д) x^2/5+2x/3≥8/15 срочно помогите пж

Answer 1

А) х^2-49>0
х^2-49=0
х^2=49
х=±7
(-∞;-7) (7;∞)
б) x^2+6x<0
x(x+6)=0
x=0 или x=-6
(0;6)
в) 2x^2-x-6≤0
2x^2-x-6=0
D=1-4×(-6)×2=49=7^2
x1=1+7/4=8/4=2
x2=1-7/8=-6/8=-3/4=-0,75
(-∞; -0,75] [2; ∞)
г) х^2-6х+9≤0
x^2-6x+9=0
D=36-4×9=0
X0=6/2=3
решений нет
д) x^2/5+2x/3≥8/15
x^2/5+2x/3-8/15=0|×15
3x^2+10x-8=0
D=100+96=196=14^2
x1=-10+14/6=4/6=2/3
x2=-10-14/6=-24/6=-4
[4;⅔]

Answer 2

Solving the Inequalities

a) To solve the inequality x^2 - 49 > 0, we can factor the left side of the inequality as (x - 7)(x + 7) > 0. This inequality holds true when either both factors are positive or both factors are negative. So, we have two cases to consider:

Case 1: (x - 7) > 0 and (x + 7) > 0 In this case, both factors are positive. Solving the inequalities, we get: x - 7 > 0 => x > 7 x + 7 > 0 => x > -7

Case 2: (x - 7) < 0 and (x + 7) < 0 In this case, both factors are negative. Solving the inequalities, we get: x - 7 < 0 => x < 7 x + 7 < 0 => x < -7

Therefore, the solution to the inequality x^2 - 49 > 0 is x < -7 or x > 7. [[1]]

b) To solve the inequality x^2 + 6x < 0, we can factor the left side of the inequality as x(x + 6) < 0. This inequality holds true when either both factors are negative or one factor is zero. So, we have two cases to consider:

Case 1: x < 0 and (x + 6) < 0 In this case, both factors are negative. Solving the inequalities, we get: x < 0 x + 6 < 0 => x < -6

Case 2: x > 0 and (x + 6) > 0 In this case, both factors are positive. Solving the inequalities, we get: x > 0 x + 6 > 0 => x > -6

Therefore, the solution to the inequality x^2 + 6x < 0 is -6 < x < 0. [[2]]

c) To solve the inequality 2x^2 - x - 6 ≤ 0, we can use the quadratic formula or factor the quadratic expression. Factoring the quadratic expression, we get (2x + 3)(x - 2) ≤ 0. This inequality holds true when either both factors are negative or both factors are positive. So, we have two cases to consider:

Case 1: (2x + 3) ≤ 0 and (x - 2) ≤ 0 In this case, both factors are negative. Solving the inequalities, we get: 2x + 3 ≤ 0 => x ≤ -3/2 x - 2 ≤ 0 => x ≤ 2

Case 2: (2x + 3) ≥ 0 and (x - 2) ≥ 0 In this case, both factors are positive. Solving the inequalities, we get: 2x + 3 ≥ 0 => x ≥ -3/2 x - 2 ≥ 0 => x ≥ 2

Therefore, the solution to the inequality 2x^2 - x - 6 ≤ 0 is -3/2 ≤ x ≤ 2. [[3]]

d) To solve the inequality x^2/5 + 2x/3 ≥ 8/15, we can multiply both sides of the inequality by 15 to eliminate the fractions. This gives us 3x^2 + 10x - 8 ≥ 0. We can then factor the quadratic expression as (3x - 2)(x + 4) ≥ 0. This inequality holds true when either both factors are negative or both factors are positive. So, we have two cases to consider:

Case 1: (3x - 2) ≥ 0 and (x + 4) ≥ 0 In this case, both factors are positive. Solving the inequalities, we get: 3x - 2 ≥ 0 => x ≥ 2/3 x + 4 ≥ 0 => x ≥ -4

Case 2: (3x - 2) ≤ 0 and (x + 4) ≤ 0 In this case, both factors are negative. Solving the inequalities, we get: 3x - 2 ≤ 0 => x ≤ 2/3 x + 4 ≤ 0 => x ≤ -4

Therefore, the solution to the inequality x^2/5 + 2x/3 ≥ 8/15 is x ≤ -4 or x ≥ 2/3. [[4]]

g) To solve the inequality x^2 - 6x + 9 ≤ 0, we can factor the left side of the inequality as (x - 3)^2 ≤ 0. A perfect square is always greater than or equal to zero, so the only solution to this inequality is when the perfect square is equal to zero. Therefore, the solution to the inequality x^2 - 6x + 9 ≤ 0 is x = 3. [[5]]

I hope this helps! Let me know if you have any further questions.