Question

Решить при всех a: а) 3(2a+x)<1-ax б) x/a<3x-1 в) x-3/a-3>1-2x/a

Answer 1

$6a + 3x < 1 - ax \\ 6a + 3x + ax - 1 = 0 \\ 7a4x - 1 = 0 \\ 7a4x = 1$

Answer 2

To solve the inequalities for all values of 'a', we'll consider each inequality separately:

а) 3(2a + x) < 1 - ax

Step 1: Distribute the 3 on the left side of the inequality. Step 2: Move all terms containing 'a' to one side and all other terms to the other side. Step 3: Factor out 'a' on the left side. Step 4: Divide both sides by the coefficient of 'a' to isolate 'a'.

Let's go through the steps:

Step 1: 6a + 3x < 1 - ax

Step 2: 6a + ax < 1 - 3x

Step 3: a(6 + x) < 1 - 3x

Step 4: a < (1 - 3x) / (6 + x)

So the solution for part (a) is: a < (1 - 3x) / (6 + x)

b) x/a < 3x - 1

Step 1: Multiply both sides by 'a' to eliminate the denominator. Step 2: Move all terms containing 'a' to one side and all other terms to the other side. Step 3: Factor out 'a' on the left side. Step 4: Divide both sides by the coefficient of 'a' to isolate 'a'.

Let's go through the steps:

Step 1: ax < 3ax - a

Step 2: ax - 3ax < -a

Step 3: a(x - 3) < -a

Step 4: a > (-a) / (x - 3)

Now, we need to be careful with the division by 'a'. Since we don't know the sign of 'a', we'll divide by 'a' and switch the inequality sign to maintain the correct direction:

a < a / (3 - x)

c) x - 3 / a - 3 > 1 - 2x / a

Step 1: Get rid of the fractions by multiplying both sides by (a - 3). Step 2: Move all terms containing 'a' to one side and all other terms to the other side. Step 3: Factor out 'a' on the left side. Step 4: Divide both sides by the coefficient of 'a' to isolate 'a'.

Let's go through the steps:

Step 1: (a - 3)(x - 3) > (1 - 2x)(a)

Step 2: ax - 3x - 3a + 9 > a - 2ax

Step 3: ax - 3x - 3a + 9 > a(1 - 2x)

Step 4: ax - 3x - 3a + 9 > a - 2ax

Step 5: ax - 3x - a > a - 2ax - 9

Step 6: ax - 3x - a + 2ax > a - 9

Step 7: 3ax - 3x - a > a - 9

Step 8: a(3x - 1) > a - 9

Now, we need to be careful with the division by 'a' again. Since we don't know the sign of 'a', we'll divide by 'a' and switch the inequality sign to maintain the correct direction:

3x - 1 > 1 - 9

Simplify further:

3x - 1 > -8

Add 1 to both sides:

3x > -7

Finally, divide by 3:

x > -7/3

So the solution for part (c) is: x > -7/3

To summarize the solutions for each part:

a) a < (1 - 3x) / (6 + x) b) a > (-a) / (3 - x) c) x > -7/3