Question

3x >-18 - 4x < -40 4(x-3)+5x< 3x2 (3x+1)-x>3 (x+4)7x+4(x-2) >6(1+3x) 7(x-3)+2 < 7x - 7​

Answer 1

3х > - 18

х > - 6

-4х < -40

х > 10

4(х - 3) + 5х < 3х

4х - 12 + 5х < 3х

6х < 12

х < 2

2(3х + 1) - х > 3(х + 4)

6х + 2 - х > 3х + 12

2х > 10

х > 5

7х + 4(х - 2) > 6(1 + 3х)

7х + 4х - 8 > 6 + 18х

7х < - 14

х < -2

7(х - 3) + 2 < 7х - 7

7х - 21 + 2 < 7х - 7

-19 < -7

х принадлежит R

Answer 2

Let's solve each inequality step by step:

1. $3x > -18$

   Divide both sides by 3 to isolate x:

   $x > -18 / 3$

   $x > -6$

2. $-4x < -40$

   Divide both sides by -4. When you divide by a negative number, reverse the inequality sign:

   $x > -40 / -4$

   $x > 10$

3. $4(x-3) + 5x < 3x^2$

   First, distribute the 4 on the left side:

   $4x - 12 + 5x < 3x^2$

   Combine like terms:

   $9x - 12 < 3x^2$

   Now, move everything to one side to set the inequality to zero:

   $3x^2 - 9x + 12 > 0$

   Now, you can solve this quadratic inequality. First, factor it:

   $(3x - 4)(x - 3) > 0$

   To determine the sign of the inequality, you can use a sign chart. Find the critical points where $3x - 4 = 0$ and $x - 3 = 0$:

   $3x - 4 = 0$ when $x = 4/3$

   $x - 3 = 0$ when $x = 3$

   Now, test the intervals created by these critical points with any value:

   • If $x < 4/3$, both factors are negative, so the expression is positive.
   • If $4/3 < x < 3$, the first factor is positive, and the second is negative, so the expression is negative.
   • If $x > 3$, both factors are positive, so the expression is positive.

   Therefore, the solution to this inequality is $x < 4/3$ or $x > 3$.

4. $3x + 1 - x > 3(x + 4)$

   First, simplify both sides:

   $2x + 1 > 3x + 12$

   Now, move everything to one side to set the inequality to zero:

   $2x - 3x + 1 - 12 > 0$

   $-x - 11 > 0$

   Multiply both sides by -1 (and reverse the inequality sign since you're multiplying by a negative number):

   $x + 11 < 0$

   Subtract 11 from both sides:

   $x < -11$

5. $7(x-3) + 2 < 7x - 7$

   First, simplify both sides:

   $7x - 21 + 2 < 7x - 7$

   Combine like terms:

   $7x - 19 < 7x - 7$

   Subtract $7x$ from both sides:

   $-19 < -7$

   This is a true statement. So, this inequality is always true and has no solution.

The solutions to the inequalities are as follows:

1. $x > -6$
2. $x > 10$
3. $x < \frac{4}{3}$ or $x > 3$
4. $x < -11$
5. No solution.