Question

Розв'язати нерівності: 1) x-9< 8x+5; 2) 4•(x-11)-5(2x-7)≥0 3) x/3 + 9 ≤ x, яке найменше ціле число є розв'язком цієї нерівності?

Answer 1

Solving Inequalities:

To solve the given inequalities, let's go through each one step by step.

1) x - 9 < 8x + 5: To solve this inequality, we need to isolate the variable x on one side of the inequality sign. Let's start by subtracting x from both sides: x - x - 9 < 8x - x + 5 -9 < 7x + 5 Next, let's subtract 5 from both sides: -9 - 5 < 7x + 5 - 5 -14 < 7x Finally, divide both sides by 7 to solve for x: -14/7 < 7x/7 -2 < x Therefore, the solution to the inequality is x > -2.

2) 4(x - 11) - 5(2x - 7) ≥ 0: To solve this inequality, let's simplify the expression first: 4x - 44 - 10x + 35 ≥ 0 -6x - 9 ≥ 0 Next, let's add 9 to both sides: -6x - 9 + 9 ≥ 0 + 9 -6x ≥ 9 Finally, divide both sides by -6. Since we are dividing by a negative number, the inequality sign will flip: -6x/(-6) ≤ 9/(-6) x ≤ -3/2 Therefore, the solution to the inequality is x ≤ -3/2.

3) x/3 + 9 ≤ x: To solve this inequality, let's start by subtracting x/3 from both sides: x/3 - x/3 + 9 ≤ x - x/3 9 ≤ 2x/3 Next, let's multiply both sides by 3 to eliminate the fraction: 9 * 3 ≤ 2x/3 * 3 27 ≤ 2x Finally, divide both sides by 2 to solve for x: 27/2 ≤ 2x/2 27/2 ≤ x Therefore, the solution to the inequality is x ≥ 27/2.

Finding the Smallest Integer Solution:

To find the smallest integer solution for the inequality x/3 + 9 ≤ x, we need to find the smallest integer value of x that satisfies the inequality.

From the previous solution, we know that x ≥ 27/2. Since we are looking for an integer solution, we need to round up the value of 27/2 to the nearest integer.

Rounding up 27/2 gives us 14. Therefore, the smallest integer solution for the inequality x/3 + 9 ≤ x is x = 14.

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