Question

3x+1<=4x-51/4-y/3>=1/3-y​

Answer 1

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Answer 2

To solve the compound inequality 3x + 1 ≤ 4x - 51/4 and 4x - 51/4 - y/3 ≥ 1/3 - y, we'll handle each inequality separately.

1. 3x + 1 ≤ 4x - 51/4: First, let's move all the x terms to one side and the constants to the other side to solve for x.

3x + 1 ≤ 4x - 51/4 3x - 4x ≤ - 51/4 - 1 -x ≤ - 55/4

Now, to isolate x, we divide both sides by -1. However, we must reverse the inequality sign when dividing by a negative number.

x ≥ 55/4

2. 4x - 51/4 - y/3 ≥ 1/3 - y: Next, we'll move all the x and y terms to one side and the constants to the other side to solve for y.

4x - 51/4 - y/3 ≥ 1/3 - y 4x - y/3 + y ≥ 1/3 + 51/4

Now, let's combine the y terms:

4x + 2y/3 ≥ 1/3 + 51/4

Next, we'll isolate y by moving the x term to the other side. Remember to change the inequality sign when moving a term to the other side.

2y/3 ≥ 1/3 + 51/4 - 4x

Now, we'll find a common denominator for the constants on the right side:

2y/3 ≥ (4 + 51)/12 - (48x)/12

2y/3 ≥ 55/12 - (48x)/12

2y/3 ≥ (55 - 48x)/12

Finally, to isolate y, we'll multiply both sides by 3/2. Remember to reverse the inequality sign when multiplying by a negative number.

y ≤ (3/2)(55 - 48x)/12

y ≤ (165 - 144x)/24

Now, we have the solutions for both inequalities:

1. x ≥ 55/4
2. y ≤ (165 - 144x)/24

These are the inequalities that satisfy the original compound inequality.