Solve the inequality. Graph the Solution.
8|x+3/4|<2
a)x<-1 or x>-1/2
b)-5/32<x<5/32
c)-1<x<-1/2
d)x<-5/32 or >5/32
8|x+3/4| < 2
8(x+3/4) < 2
8x+6 < 2
8x < 2-6
8x < -4
X < -1/2.
8|-x-3/4| < 2
8(-x-3/4) < 2
-8x-6 < 2
-8x < 2+6
-8x < 8
X > -1
Solution Set: -1< X < -1/2.
To solve the inequality 8|x + 3/4| < 2, we need to isolate the absolute value term.
1. Divide both sides of the inequality by 8:
|x + 3/4| < 2/8
Simplifying, we get:
|x + 3/4| < 1/4
2. Since the absolute value is less than a positive number, we can split the inequality into two cases:
Case 1: x + 3/4 < 1/4
Solving for x:
x < 1/4 - 3/4
x < -2/4
x < -1/2
Case 2: -(x + 3/4) < 1/4
Solving for x:
-x - 3/4 < 1/4
-x < 1/4 + 3/4
-x < 4/4
-x < 1
Dividing by -1 (which changes the inequality sign):
x > -1
3. Combining the results from the two cases, we have:
x < -1/2 or x > -1
Therefore, the correct answer is option a) x < -1 or x > -1/2.
To graph the solution on a number line, you will plot an open circle on -1/2 (since it is not included) and shade the regions to the left of -1/2 and to the right of -1.
To solve the inequality 8|x+3/4| < 2, we can follow these steps:
Step 1: Isolate the absolute value expression.
Divide both sides of the inequality by 8:
|x + 3/4| < 2/8
Simplify:
|x + 3/4| < 1/4
Step 2: Apply the definition of absolute value.
|x + 3/4| is less than 1/4, which means the distance between x + 3/4 and 0 is less than 1/4.
Step 3: Set up two cases.
Case 1: x + 3/4 > 0
For this case, we have x + 3/4 < 1/4. Solve for x:
x + 3/4 < 1/4
x < 1/4 - 3/4
x < -2/4
Simplifying, we get:
x < -1/2
Case 2: x + 3/4 < 0
For this case, we have -(x + 3/4) < 1/4. Solve for x:
-(x + 3/4) < 1/4
-x - 3/4 < 1/4
-x < 1/4 + 3/4
-x < 4/4
Simplifying, we get:
-x < 1
Multiplying both sides by -1, we have to change the inequality direction:
x > -1
Step 4: Combine the solutions from both cases.
The solutions for x are x < -1/2 and x > -1. This can be represented using interval notation as (-∞, -1/2) U (-1, +∞).
Step 5: Graph the solution.
On a number line, we mark -1/2 with an open circle, indicating that it is not included in the solution. Then we mark -1 with an open circle since it is also not included in the solution. Finally, we shade the number line to the left of -1/2 and to the right of -1, representing all the values that satisfy the inequality. The resulting graph should show the shaded regions with open circles at -1/2 and -1.
Therefore, the correct answer is option a) x < -1 or x > -1/2.