Answer:
To solve the given inequality, let's break it down step by step:
1. Simplify the expression within the inequality:
1 - 4 = -3
2 + 1 = 3
3 * 1 = 3
Therefore, the inequality becomes: -3 < 3 * 20.
2. Simplify further:
3 * 20 = 60
The inequality now becomes: -3 < 60.
3. Graphing the solution:
To graph the solution, we need to represent all values that satisfy the inequality on a number line. Since -3 is less than 60, we can shade the region to the right of -3 and extend it up to positive infinity (∞). This indicates that any value greater than -3 satisfies the inequality.
[number line with a shaded region starting from -3 and extending to ∞]
4. Interval notation:
In interval notation, we represent the solution as (-∞, ∞), which means all real numbers.
5. Testing values:
To verify our solution, we can choose test values within and outside the shaded region and substitute them into the original inequality. If the inequality holds true for these values, then our solution is correct.
Let's choose three test values: -4, 0, and 100.
For x = -4:
Substitute x = -4 into the original inequality:
1 - 4(2 + 1) * (20) < 0
-3 * (20) < 0
-60 < 0
Since -60 is less than zero, this test value satisfies the inequality.
For x = 0:
Substitute x = 0 into the original inequality:
1 - 4(2 + 1) * (20) < 0
-3 * (20) < 0
-60 < 0
Again, -60 is less than zero, so this test value satisfies the inequality.
For x = 100:
Substitute x = 100 into the original inequality:
1 - 4(2 + 1) * (20) < 0
-3 * (20) < 0
-60 < 0
Once more, -60 is less than zero, so this test value satisfies the inequality.
Therefore, all three test values confirm that our solution (-∞, ∞) is correct.
Explanation: