Question

|х+3|+|х+5|≤10 Help me please

Answer 1

Ответ:

[-9;1]

Объяснение:

Answer 2

To solve the inequality |x + 3| + |x + 5| ≤ 10, we'll need to consider different cases based on the values of x that make the absolute value expressions positive or negative. Let's break it down step by step:

1. When x + 3 is non-negative (x + 3 ≥ 0), and x + 5 is non-negative (x + 5 ≥ 0): In this case, both absolute value expressions are positive, so we can rewrite the inequality as: x + 3 + x + 5 ≤ 10 Simplify: 2x + 8 ≤ 10 Subtract 8 from both sides: 2x ≤ 2 Divide by 2 (since 2 is positive): x ≤ 1

2. When x + 3 is non-negative (x + 3 ≥ 0), and x + 5 is negative (x + 5 < 0): In this case, the absolute value of x + 5 is the negative of its actual value, so the inequality becomes: x + 3 - (x + 5) ≤ 10 Simplify: x + 3 - x - 5 ≤ 10 -2 ≤ 10 This is always true.

3. When x + 3 is negative (x + 3 < 0), and x + 5 is non-negative (x + 5 ≥ 0): Similar to the previous case, the absolute value of x + 3 is the negative of its actual value. So the inequality becomes: -(x + 3) + (x + 5) ≤ 10 Simplify: -x - 3 + x + 5 ≤ 10 2 ≤ 10 This is always true.

4. When x + 3 is negative (x + 3 < 0), and x + 5 is negative (x + 5 < 0): In this case, both absolute value expressions are negative, so we can rewrite the inequality as: -(x + 3) - (x + 5) ≤ 10 Simplify: -x - 3 - x - 5 ≤ 10 -2x - 8 ≤ 10 Divide by -2 (since -2 is negative): x + 4 ≥ -5 Subtract 4 from both sides: x ≥ -9

Now, let's summarize the solutions from each case:

1. x ≤ 1
2. Always true
3. Always true
4. x ≥ -9

So, the solutions for the inequality |x + 3| + |x + 5| ≤ 10 are: -9 ≤ x ≤ 1