Question

Решите |x + 1 | > | 3 - 2x |

Answer 1

!!!!!!!!!!!!!!!!!!!!!!!!!!!

Answer 2

To solve the inequality |x + 1| > |3 - 2x|, we'll need to consider different cases depending on the value of x. The absolute value of a number can be either positive or zero, so we'll need to account for both possibilities.

Case 1: (x + 1) and (3 - 2x) are both positive In this case, the inequality remains the same since the absolute values will not affect the direction of the inequality: x + 1 > 3 - 2x

Simplify the inequality: x + 2x > 3 - 1 3x > 2

Case 2: (x + 1) and (3 - 2x) are both negative When both expressions inside the absolute values are negative, the absolute values effectively change the direction of the inequality: -(x + 1) > -(3 - 2x)

Simplify the inequality: -x - 1 > -3 + 2x

Bring all x terms to one side of the inequality: -x - 2x > -3 + 1 -3x > -2

Now, we need to consider the case when one expression inside the absolute value is positive, and the other is negative:

Case 3: (x + 1) is positive and (3 - 2x) is negative When x + 1 is positive, but 3 - 2x is negative, the absolute value |x + 1| becomes (x + 1), and |3 - 2x| becomes -(3 - 2x):

x + 1 > -(3 - 2x)

Simplify the inequality: x + 1 > -3 + 2x

Move all x terms to one side of the inequality: x - 2x > -3 - 1 -x > -4

Now, let's consider the last case:

Case 4: (x + 1) is negative and (3 - 2x) is positive When x + 1 is negative, but 3 - 2x is positive, the absolute value |x + 1| becomes -(x + 1), and |3 - 2x| becomes (3 - 2x):

-(x + 1) > 3 - 2x

Simplify the inequality: -x - 1 > 3 - 2x

Move all x terms to one side of the inequality: -x + 2x > 3 + 1 x > 4

Now, we have four cases, and we need to combine their results. Remember that we have to consider both the sign and the magnitude of the results:

Case 1: 3x > 2 Case 2: -3x > -2 (Remember to flip the inequality when multiplying by a negative number) Case 3: x > -4 + 2x Case 4: x > 4

Now, let's simplify Case 3: x - 2x > -4 -x > -4

To combine Case 3 and Case 2, we need to flip the inequality of Case 2 and add them:

-3x > -2 and -x > -4

Combine the inequalities:

-3x + x > -2 + (-4) -2x > -6

Now, divide by -2, and remember to reverse the inequality when dividing by a negative number:

x < 3

So, the final solution for the inequality |x + 1| > |3 - 2x| is: x > 4 or x < 3