Solve the inequality. Graph the solution.
|2x+7|≤27
(1 point)
Responses
To solve the inequality |2x+7| ≤ 27, we need to consider two cases: when 2x+7 is positive and when it is negative.
Case 1: 2x+7 ≥ 0
In this case, the absolute value of 2x+7 is equal to 2x+7 itself. So we have 2x+7 ≤ 27.
Subtracting 7 from both sides, we get 2x ≤ 20.
Dividing both sides by 2, we have x ≤ 10.
Case 2: 2x+7 < 0
In this case, the absolute value of 2x+7 is equal to -(2x+7), so we have -(2x+7) ≤ 27.
Multiplying both sides by -1 (which reverses the inequality), we get 2x+7 ≥ -27.
Subtracting 7 from both sides, we have 2x ≥ -34.
Dividing both sides by 2, we get x ≥ -17.
Combining the solutions from both cases, we have -17 ≤ x ≤ 10.
To graph the solution, we plot a number line with a closed circle at -17 and a closed circle at 10, and shade the area in between the two points, including the points themselves.
Here is a rough sketch of the graph:
<----[-17]=====================[10]---->
To solve the inequality |2x+7| ≤ 27, we need to consider two cases: when 2x+7 is positive and when it is negative.
Case 1: 2x+7 ≥ 0
In this case, we have |2x+7| = 2x+7, so the inequality becomes 2x+7 ≤ 27.
Solving this inequality, we subtract 7 from both sides:
2x ≤ 27 - 7
2x ≤ 20
Dividing both sides by 2:
x ≤ 10
Case 2: 2x+7 < 0
In this case, we have |2x+7| = -(2x+7), so the inequality becomes -(2x+7) ≤ 27.
To simplify this, we multiply both sides by -1, which changes the direction of the inequality:
2x+7 ≥ -27
Subtracting 7 from both sides:
2x ≥ -27 - 7
2x ≥ -34
Dividing both sides by 2:
x ≥ -17
Combining the two cases, we have the solution x ≤ 10 and x ≥ -17.
To graph the solution, we need to plot the numbers on a number line. We mark a closed circle at 10 and an open circle at -17. Then, we shade the region between these two points since both x ≤ 10 and x ≥ -17 are true.
Here is the graph of the solution:
-17 10
-----------o--------------o-----
| shaded region |