Question

1] 3x>-9. 2] -0,3x<9,6. 3] 2x+18>0. 4] x+5⩽2x-3. 5] 2(x+3)⩾2x. 6] 3(x+2)-2(5-x)<1. 7] 6x-3(0,2x+3)<2x. 8] (x-2)(x+1)>0. 9] (x+5)(x+7)⩽0. 9] x²-9>0. 10] 3x²⩽12x. 11] x²-9x<0

Answer 1

$1)\; \; 3x>-9\; \; ,\; \; x>-3\\\\2)\; \; -0,3x-32\\\\3)\; \; 2x+18>0\; \; ,\; \; x>-9\\\\4)\; \; x+5\leq 2x-3\; \; ,\; \; x\geq 8\\\\5)\; \; 2(x+3)\geq 2x\; \; ,\; \; 2x+6\geq 2x\; \; ,\; \; 6\geq 0\; \; \to \; \; x\in R\\\\6)\; \; 3(x+2)-2(5-x)$

$8)\; \; (x-2)(x+1)>0\; \; ,\; \; x\in (-\infty ,-1)\cup (2,+\infty )\\\\9)\; \; (x+5)(x+7)\leq 0\; \; ,\; \; x\in [-7,-5\, ]\\\\10)\; \; x^2-9>0\; \; ,\; \; (x-3)(x+3)>0\; ,\; x\in (-\infty ,-3)\cup (3,+\infty )\\\\11)\; \; 3x^2\leq 12x\; \; ,\; \; 3x(x-4)\leq 0\; ,\; \; x\in [\, 0,4\, ]\\\\12)\; \; x^2-9x$

Answer 2

Let's solve each of the inequalities one by one:

1. 3x > -9: To solve this inequality, we divide both sides by 3. Since we're dividing by a positive number, the inequality sign remains the same.

Dividing both sides by 3: x > -3

Therefore, the solution to the inequality is x > -3.

2. -0.3x < 9.6: To solve this inequality, we divide both sides by -0.3. When we divide by a negative number, the inequality sign flips.

Dividing both sides by -0.3: x > 9.6 / -0.3

Simplifying the right side: x > -32

So, the solution to the inequality is x > -32.

3. 2x + 18 > 0: To solve this inequality, we subtract 18 from both sides:

2x > -18

Dividing both sides by 2: x > -9

The solution to the inequality is x > -9.

4. x + 5 ≤ 2x - 3: To solve this inequality, let's isolate the x terms on one side:

x - 2x ≤ -3 - 5

Simplifying both sides: -x ≤ -8

Dividing both sides by -1 (and flipping the inequality sign): x ≥ 8

The solution to the inequality is x ≥ 8.

5. 2(x + 3) ≥ 2x: Expanding the left side: 2x + 6 ≥ 2x

The inequality simplifies to 6 ≥ 0, which is always true.

Therefore, the solution to the inequality is all real numbers.

6. 3(x + 2) - 2(5 - x) < 1: Expanding both sides: 3x + 6 - 10 + 2x < 1

Combining like terms: 5x - 4 < 1

Adding 4 to both sides: 5x < 5

Dividing both sides by 5: x < 1

The solution to the inequality is x < 1.

7. 6x - 3(0.2x + 3) < 2x: Distributing -3 to the terms inside the parentheses: 6x - 0.6x - 9 < 2x

Combining like terms: 5.4x - 9 < 2x

Adding 9 to both sides: 5.4x < 2x + 9

Subtracting 2x from both sides: 3.4x < 9

Dividing both sides by 3.4: x < 9 / 3.4

Simplifying the right side: x < 2.647

The solution to the inequality is x < 2.647.

8. (x - 2)(x + 1) > 0: To solve this inequality, we examine the signs of the expressions (x - 2) and (x + 1) and determine when their product is positive.

Case 1: Both expressions are positive. x - 2 > 0 and x + 1 > 0 x > 2 and x > -1

The solution is x > 2, as it satisfies both conditions.

Case 2: Both expressions are negative. x - 2 < 0 and x + 1 < 0 x < 2 and x < -1

There is no solution in this case.

Case 3: One expression is positive and the other is negative. x - 2 > 0 and x + 1 < 0 x > 2 and x < -1

There is no solution in this case either.

Therefore, the solution to the inequality is x > 2.

9. (x + 5)(x + 7) ≤ 0: To solve this inequality, we examine the signs of the expressions (x + 5) and (x + 7) and determine when their product is less than or equal to zero.

Case 1: Both expressions are positive or zero. x + 5 ≥ 0 and x + 7 ≥ 0 x ≥ -5 and x ≥ -7

The solution is x ≥ -5, as it satisfies both conditions.

Case 2: Both expressions are negative. x + 5 < 0 and x + 7 < 0 x < -5 and x < -7

The solution is x < -7, as it satisfies both conditions.

Case 3: One expression is positive or zero, and the other is negative. x + 5 ≥ 0 and x + 7 < 0 x ≥ -5 and x < -7

The solution is -7 < x ≤ -5.

Therefore, the solution to the inequality is -7 < x ≤ -5.

10. x² - 9 > 0: To solve this inequality, we factor the quadratic expression:

(x - 3)(x + 3) > 0

We examine the signs of the expressions (x - 3) and (x + 3) to determine when their product is greater than zero.

Case 1: Both expressions are positive. x - 3 > 0 and x + 3 > 0 x > 3 and x > -3

The solution is x > 3, as it satisfies both conditions.

Case 2: Both expressions are negative. x - 3 < 0 and x + 3 < 0 x < 3 and x < -3

The solution is x < -3, as it satisfies both conditions.

Case 3: One expression is positive and the other is negative. x - 3 > 0 and x + 3 < 0 x > 3 and x < -3

There is no solution in this case.

Therefore, the solution to the inequality is x < -3 or x > 3.

11. x² - 9x < 0: To solve this inequality, we factor out x:

x(x - 9) < 0

We examine the signs of the expressions x and (x - 9) to determine when their product is less than zero.

Case 1: Both expressions are positive. x > 0 and x - 9 > 0 x > 0 and x > 9

The solution is x > 9, as it satisfies both conditions.

Case 2: Both expressions are negative. x < 0 and x - 9 < 0 x < 0 and x < 9

The solution is x < 0, as it satisfies both conditions.

Case 3: One expression is positive and the other is negative. x > 0 and x - 9 < 0 x > 0 and x < 9

The solution is 0 < x < 9.

Therefore, the solution to the inequality is 0 < x < 9.