Questionc. $4 b^{2}+14 b+16<10$

Studdy Solution

STEP 1

What is this asking? Find the values of $b$ that make the expression $4b^2 + 14b + 16$ less than 10. Watch out! Don't forget to flip the inequality sign when multiplying or dividing by a negative number!

STEP 2

1. Rewrite the inequality
2. Factor the quadratic
3. Find the roots
4. Determine the solution interval

STEP 3

Let's start by rewriting the inequality so that one side is zero.
We can do this by subtracting 10 from both sides: $4b^2 + 14b + 16 - 10 < 0$ $4b^2 + 14b + 6 < 0$

STEP 4

To make things a little easier, we can divide both sides of the inequality by 2.
Since 2 is positive, the inequality sign stays the same: $2b^2 + 7b + 3 < 0$

STEP 5

Now, we want to factor our quadratic expression!
We're looking for two numbers that multiply to $(2 \cdot 3 = 6)$ and add up to 7.
Those numbers are 6 and 1! $2b^2 + 6b + b + 3 < 0$

STEP 6

Let's factor by grouping: $2b(b+3) + 1(b+3) < 0$ $(2b+1)(b+3) < 0$

STEP 7

To find the roots, or the values of $b$ that make the expression equal to zero, we set each factor equal to zero and solve for $b$ : $2b + 1 = 0$ $2b = -1$ $b = -\frac{1}{2}$ And: $b + 3 = 0$ $b = -3$ So, our roots are $b = -\frac{1}{2}$ and $b = -3$ .

STEP 8

Let's visualize this on a number line!
We have our two roots, -3 and -1/2.
We want to know when the expression $(2b+1)(b+3)$ is less than zero.

STEP 9

If $b < -3$ , both factors are negative, so their product is positive.
If $b$ is between -3 and -1/2, one factor is negative and the other is positive, making the product negative.
If $b > -\frac{1}{2}$ , both factors are positive, so their product is positive.

STEP 10

We want the interval where the product is less than zero, which is between our two roots.
So, our solution is: $-3 < b < -\frac{1}{2}$

STEP 11

The solution to the inequality $4b^2 + 14b + 16 < 10$ is $-3 < b < -\frac{1}{2}$ .

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