How do you graph and solve #| 3x + 8 | +4

Question

How do you graph and solve #| 3x + 8 | +4 <=0#?

Isaiah Anthony · Accepted Answer

See a solution process below:

First, subtract #color(red)(4)# from each side of the inequality to isolate the absolute value function while keeping the inequality balanced: #abs(3x + 8) + 4 - color(red)(4) <= 0 - color(red)(4)# #abs(3x + 8) + 0 <= -4# #abs(3x + 8) <= -4# However, the absolute value function takes any number and converts it to #0# or it's positive form. Therefore, the output of the absolute value function will always be #>= 0#. So, the absolute value of #(3x + 8)# cannot be less than a negative number. So, there is no answer to this problem. Or, the answer is the null or empty set: #{O/}#

Isaiah Anthony · Answer

undefined To graph and solve the inequality |3x + 8| + 4 ≤ 0:

Step 1: Rewrite the absolute value inequality as two separate inequalities:

1. 3x + 8 + 4 ≤ 0
2. -(3x + 8) + 4 ≤ 0

Step 2: Solve each inequality separately:

For the first inequality:
3x + 12 ≤ 0
3x ≤ -12
x ≤ -4

For the second inequality:
-3x - 8 + 4 ≤ 0
-3x - 4 ≤ 0
-3x ≤ 4
x ≥ -4/3

Step 3: Combine the solutions:
x ≤ -4 and x ≥ -4/3

Step 4: Graph the solution set on a number line:

[______________________●-----------------------●_____________________]

Step 5: Shade the regions that satisfy the inequality:
Shade the region to the left of -4 and the region to the right of -4/3.

Step 6: Combine the shaded regions:
The solution set is the interval from -∞ to -4, including -4, and the interval from -4/3 to +∞.

Step 7: Write the final solution:
x ∈ (-∞, -4] ∪ [-4/3, +∞)

Eva Adamson · Answer

undefined To graph and solve the inequality $|3x + 8| + 4 \leq 0$, follow these steps:

1. Subtract 4 from both sides of the inequality:
$$|3x + 8| \leq -4$$

2. Absolute values cannot be negative, so the expression inside the absolute value must be either zero or negative:
$$3x + 8 \leq 0$$

3. Solve the inequality $3x + 8 \leq 0$ for $x$:
$$3x \leq -8$$
$$x \leq -\frac{8}{3}$$

4. The solution to the inequality $|3x + 8| + 4 \leq 0$ is $x \leq -\frac{8}{3}$.

5. To graph the solution on a number line, plot an open circle at $-\frac{8}{3}$ and shade the region to the left of this point, including negative infinity. This represents all the values of $x$ that satisfy the inequality.

So, the solution and graph for the inequality $|3x + 8| + 4 \leq 0$ are:

Solution: $x \leq -\frac{8}{3}$

Graph:

```
-----------------------------●--->
        -8/3
```

The shaded region represents the solution set, which includes all $x$ values less than or equal to $-\frac{8}{3}$.

How do you graph and solve #| 3x + 8 | +4 &lt;=0#?

How do you graph and solve #| 3x + 8 | +4 <=0#?