The3Mathketeers

The given information states that 4 pencils is sold for every 5 pens. This information can be represented mathematically as a ratio \(\frac{4 \text{ pencils}}{5 \text{ pens}}\). We know that the store sold 28 pencils yesterday. We can set up a proportion and solve to determine the number of pens sold.

\(\frac{4\text{ pencils}}{5\text{ pens}} = \frac{28\text{ pencils}}{p \text{ pens}} \\ 4p = 28 * 5 \\ p = \frac{28 * 5}{4} \\ p = 7 * 5 \\ p = 35 \text{ pens}\)

Let \(M_1\) and \(M_2\) be arbitrary matrices with \(r\) rows and \(c\) columns. The operation of addition or subtraction of matrices is defined if and only if \(r_{M_1} = r_{M_2}\) and \(c_{M_1} = c_{M_2}\). I will do the first few, and you should be able to do the rest.

a) \(L - N\)

\(L\) is a 2x2 matrix, which means that L has 2 rows and 2 columns. N is a 4x2 matrix, which means that it has 4 rows and 2 columns. The numbers of rows is different. Therefore, \(L-N\) is not a defined operation.

b) \(M-N\)

\(M\) is a 4x2 matrix, which means that M has 4 rows and 2 columns. N is a 4x2 matrix, which means that it has 4 rows and 2 columns. The number of rows and the number of columns are the same. Therefore, \(M - N\) is a defined operation.

You should be able to complete the next two. Let me know if you have any questions about it.

Usually, finding the x-intercepts of a cubic function is quite challenging, but some observation reveals we can use factoring to find the x-intercepts of this particular cubic function \(f(x) = -3x^3 - 6x^2 + 3x + 6\) by setting the function equal to 0 and then solving for the individual x-values.

\(-3x^3 - 6x^2 + 3x + 6 = 0 \\ -3(x^3 + 2x^2 -x - 2 = 0 \\ -3[x^2(x + 2) - 1(x + 2)] = 0 \\ -3(x^2 - 1)(x + 2) = 0 \\ -3(x - 1)(x + 1)(x + 2) = 0 \\ x=1 \text{ or } x = -1 \text{ or } x = -2\)

These x-values indicate the x-coordinate of the x-intercept. The y-coordinate of the x-intercept is always 0. Therefore, the x-intercepts are \((1, 0), (-1, 0), \text{ and } (-2, 0)\).

With this information of the x-intercept and the negative nature of the leading coefficient, you should be able to make a reasonably accurate graph of this cubic function, but below is one for reference.

Read the definition of the function and apply it to each input.

\(f(i) = i + 1 \\ f(f(1)) = f(1 - i) = 2 - i \\ f(f(f(-1))) = f(f(-1 - i)) = f(-i) = -i + 1 \\ f(f(f(f(-i)))) = f(f(f(-i + 1))) = f(f(-i + 2)) = f(-i + 3) = -i + 4\)

Now, find its sum.

\(\begin{align*} f(i) + f(f(1)) + f(f(f(-1))) + f(f(f(f(-i)))) &= i + 1 + 2 - i - i + 1 -i + 4 \\ &= -2i + 8 \end{align*}\)

Greetings, SpectraSynth!

You made a great attempt to solve this inequality, but I noticed you made an algebraic error along the way. You subtracted the quantity \(6x-2\) incorrectly from both sides of the inequality, which lead your answer astray.

If you manipulate the inequality by completing the square, the x-values that satisfy and do not satisfy this inequality become clear rather quickly.

\(2x^2 + 8x < -6 + 6x + 4 \\ 2x^2 + 8x < 6x - 2 \\ 2x^2 + 2x < -2 \\ 2(x^2 + x) < -2 \\ 2\left(x^2 + x + \frac{1}{4}\right) < -2 + 2 * \frac{1}{4} \\ 2\left(x + \frac{1}{2}\right)^2 < -\frac{3}{2} \)

Now, analyze the inequality carefully. I noticed that the left-hand side is the product of two nonnegative quantities. 2 is positive, and \(\left(x + \frac{1}{2}\right)^2\) has a square, which guarantees that the result is either 0 or positive in nature. If we multiply 2 by a quantity that is either 0 or positive, then it is impossible for that quantity to be less than \(-\frac{3}{2}\). This indicates that there is no solution to this inequality. In interval notation, that is typically denoted as \(\emptyset\).

maximum, this is a good-faith effort towards solving this problem, but you forgot to consider the right-hand side of the equation. Also, your attempt at factorization is not correct.

If you check your work, you will discover that your claims are not accurate.\(2(x + 10)(x + 3) = 2(x^2 + 10x + 3x + 30) = 2(x^2 + 13x + 30) \neq 2(x^2 + 12x + 30)\).

SpectraSynth, this is a good-faith effort towards solving this problem, but why did the line \(2x^2 + 24x - 60 - x^2 - 13x = 0\) turn into \(x^2 + 4x - 60 = 0\)? Notice how \(24x - 13x \neq 4x\).

Let's do some algebraic manipulation and see what the result yields.

\(2x^2 + 24x - 60 = x(x + 13) \\ 2x^2 +24x - 60 = x^2 + 13x \\ x^2 + 11x - 60 = 0\)

This quadratic happens to be factorable, so I will take advantage of that fact.

\((x - 4)(x+ 15) = 0 \\ x = 4 \text{ or } x = -15\)

Now, just pick the smallest x-value, which is \(x = -15\) in this case.