Question

|3x+5|>20 2)|7-4x|<11 3)|4+3x|<5

Answer 1

$1)\; \; |3x+5|>20\; \; \to \; \; \; \left [ {{3x+5>20} \atop {3x+515} \atop {3x5} \atop {x$

$2)\; \; |7-4x|$

$3)\; \; |4+3x|$

Answer 2

To solve absolute value inequalities, we need to consider two cases: when the expression inside the absolute value is positive and when it is negative. Let's solve each of the given inequalities step by step.

1. |3x + 5| > 20:

Case 1: (3x + 5) > 20 3x > 20 - 5 3x > 15 x > 15/3 x > 5

Case 2: -(3x + 5) > 20 -3x - 5 > 20 -3x > 20 + 5 -3x > 25 Divide both sides by -3, but remember to flip the inequality sign: x < 25/-3 x < -25/3

So the solution to |3x + 5| > 20 is x < -25/3 or x > 5.

2. |7 - 4x| < 11:

Case 1: (7 - 4x) < 11 -4x < 11 - 7 -4x < 4 Divide both sides by -4, but remember to flip the inequality sign: x > 4/-4 x > -1

Case 2: -(7 - 4x) < 11 -7 + 4x < 11 4x < 11 + 7 4x < 18 x < 18/4 x < 9/2

So the solution to |7 - 4x| < 11 is -1 < x < 9/2.

3. |4 + 3x| < 5:

Case 1: (4 + 3x) < 5 3x < 5 - 4 3x < 1 Divide both sides by 3: x < 1/3

Case 2: -(4 + 3x) < 5 -4 - 3x < 5 -3x < 5 + 4 -3x < 9 Divide both sides by -3, but remember to flip the inequality sign: x > 9/-3 x > -3

So the solution to |4 + 3x| < 5 is -3 < x < 1/3.

In summary:

1. |3x + 5| > 20: x < -25/3 or x > 5.
2. |7 - 4x| < 11: -1 < x < 9/2.
3. |4 + 3x| < 5: -3 < x < 1/3.