Solve each inequality. State the solution set using interval notation and graph the solution set.
1-x<6 or -5+x<1
1-x<6:
To solve the inequality, we need to isolate the variable x.
1 - x < 6
-x < 6 - 1
-x < 5
When we multiply or divide both sides of an inequality by a negative number, the inequality sign switches. So, let's multiply both sides of the inequality by -1.
(-1)(-x) > 5(-1)
x > -5
The solution to the inequality 1 - x < 6 is x > -5.
Next, let's solve the second inequality:
-5 + x < 1:
To solve the inequality, we need to isolate the variable x.
-5 + x < 1
x < 1 + 5
x < 6
The solution to the inequality -5 + x < 1 is x < 6.
Now, let's state the solution sets using interval notation:
For x > -5, the solution set is (-5, ∞).
For x < 6, the solution set is (-∞, 6).
Finally, let's graph the solution set on a number line:
-∞ -5 6 ∞
-------------------●-----●-------------------
The shaded region to the right of -5 and the left of 6 represents the solution set to the given inequalities.
To solve each inequality, we will isolate the variable and determine the solution set for each inequality separately.
1. Solving the first inequality, 1 - x < 6:
To isolate the variable x, we need to get rid of the constant on the left side by subtracting 1 from both sides:
1 - x - 1 < 6 - 1
-x < 5
Next, we divide both sides by -1. Since we divide by a negative number, we must reverse the inequality sign:
(-x)/(-1) > 5/(-1)
x > -5
Therefore, the solution to the first inequality is x > -5.
2. Solving the second inequality, -5 + x < 1:
To isolate the variable x, we need to get rid of the constant on the left side by adding 5 to both sides:
-5 + x + 5 < 1 + 5
x < 6
Therefore, the solution to the second inequality is x < 6.
Now, let's state the solution set using interval notation and graph the solution on a number line:
For the first inequality, x > -5, we write the interval notation as (-5, ∞), which represents all the numbers greater than -5.
For the second inequality, x < 6, we write the interval notation as (-∞, 6), which represents all the numbers less than 6.
Graphically, on a number line, we represent the solution set by shading the regions between -5 and 6, excluding the endpoints. The shaded region represents all values that satisfy both inequalities.