Final answer:
The inequality |3x+4|<8 indicates two scenarios, leading to solutions where x < 4/3 or x > -4. On a number line, this is represented by shading the region between open circles at x = 4/3 and x = -4.
Step-by-step explanation:
The inequality |3x+4|<8 is a mathematical expression that requires us to consider two separate cases because of the absolute value. The solution set of this inequality represents all the x-values that make the inequality true. To find these values, we split the inequality into two scenarios: one where 3x+4 is positive and one where it is negative.
1. If 3x+4 is positive, then we simply remove the absolute value to get 3x+4 < 8. To find x, subtract 4 from both sides to get 3x < 4, and then divide by 3 to find that x < 4/3.
2. If 3x+4 is negative, then we must consider the inequality as -(3x+4) < 8. To solve for x, first distribute the negative sign to get -3x-4 < 8. Then add 4 to both sides to obtain -3x < 12, and finally, divide by -3, remembering to reverse the inequality sign, giving x > -4.
To graph the solution on a number line, we draw an open circle at x = 4/3 and another at x = -4, and shade the region between these points. The open circles indicate that the points x = 4/3 and x = -4 are not included in the solution set.