We got two solutions, b≈ 0.54 and b≈ 7.46. Let's plot these on a number line. Since the inequality is strict, these solutions should not be included in the solution set, so we will draw the points open.
Now we are going to test some points to find where we will draw our lines. One of our test points will be less than 0.54, a second will be between 0.54 and 7.46, and a third will be greater than 7.46. We will shade the number line according to which values give a true inequality. Let's use 0, 4, and 8.
b
(b-4)^2 < 12
Evaluate
True?
0
(0-4)^2 ? < 12
16 ≮ 12
*
4
(4-4)^2 ? < 12
0 < 12
✓
8
(8-4)^2 ? < 12
16 ≮ 12
*
The middle value produced a true statement. This means that wWhen b is between 0.54 and 7.46, the inequality is true.
b Like we did in Part A, we will solve this inequality by first treating it as an equation and solving for x.
We got two solutions, x=-1 and x=-5. Let's plot these on a number line. Since the inequality is strict, they should not be included in the solution set and therefore, we make them open.
Let's test three points, one less than -5, the second between -5 and -1, and the third above -1. Let's use -6, - 3, and 0.
x
(x+3)^2 > 4
Evaluate
True?
-6
(-6 +3)^2 ? > 4
9 > 4
✓
-3
(-3 +3)^2 ? > 4
0 ≯ 4
*
0
(0 +3)^2 ? > 4
9 > 4
✓
When x is less than -5 or above -1, the inequality is true, so we will color in both ends of the number line.
c Notice that this inequality does not square the variable. This means that we do not have to treat the expression as an equality first — we can get right to solving it.
When x is greater than 5, the inequality is true. Like in Parts A and B, we have a strict inequality, which means the endpoint is open. Let's draw our number line.
