Question 1

Solve the problem.
-Alan is building a garden shaped like a rectangle with a semicircle attached to one short side. If he has 40 feet of fencing to go around it, what dimensions will give him the maximum area in the garden?

Accepted Answer

The perimeter constraint is $2l + w + \frac{\pi w}{2} = 40$. Solving for maximum area, the width is $\frac{80}{\pi + 4}$ and length is half of that width.

Question 2

Solve the inequality.
-$$x ^ { 2 } - 4 x - 5 \leq 0$$

Accepted Answer

To solve the inequality $x^2 - 4x - 5 \leq 0$, first factor the quadratic equation. It factors to $(x - 5)(x + 1) \leq 0$. The roots are $x = 5$ and $x = -1$. The inequality is satisfied between these two roots, including the endpoints, hence the solution is $[ - 1,5 ]$.

Question 3

Solve the inequality.
-$$x ^ { 2 } + 4 x \geq 0$$

Accepted Answer

The inequality can be factored as $x(x + 4) \geq 0$, which is true for $x \leq -4$ or $x \geq 0$.

Question 4

Solve the problem.
-If $h ( x ) = x ^ { 2 } + 2 x - 8$, solve $h ( x ) > 0$.

Accepted Answer

To solve $h(x) > 0$, we first find the roots of the quadratic equation $x^2 + 2x - 8 = 0$. Factoring or using the quadratic formula, we find the roots to be $x = -4$ and $x = 2$. The parabola opens upwards (since the coefficient of $x^2$ is positive), so $h(x) > 0$ outside the interval between the roots. Therefore, the solution is $(-\infty, -4)$ or $(2, \infty)$.

Question 5

Solve the problem.
-If $f ( x ) = 6 x ^ { 2 } - 5 x$ and $g ( x ) = 2 x + 3$, solve $f ( x ) \leq g ( x )$.

Accepted Answer

The answer of Solve the problem.
-If \(f ( x )...

Question 6

Solve the problem.
-If $f ( x ) = 6 x ^ { 2 } - 5 x$ and $g ( x ) = 2 x + 3$, solve for $f ( x ) = g ( x )$

Accepted Answer

To solve for $f(x) = g(x)$, set $6x^2 - 5x = 2x + 3$ and solve for $x$. Rearranging gives $6x^2 - 7x - 3 = 0$. Factoring or using the quadratic formula yields $x = \frac{3}{2}$ or $x = -\frac{1}{3}$, which matches choice B.

Question 7

Solve the inequality.
-$$x ^ { 2 } - 36 \leq 0$$

Accepted Answer

The inequality $x^2 - 36 \leq 0$ simplifies to $(x - 6)(x + 6) \leq 0$, which means $x$ is between -6 and 6, inclusive. Thus, the correct interval is $[-6, 6]$.

Question 8

Solve the inequality.
-$64 x ^ { 2 } + 9 < 48 x$

Accepted Answer

The answer of Solve the inequality.
-\(64 x ^ { 2...

Question 9

Solve the inequality.
-$$x ^ { 2 } - 7 x \geq - 12$$

Accepted Answer

First, rearrange the inequality to standard form: $x^2 - 7x + 12 \geq 0$. Factoring the quadratic gives $(x - 3)(x - 4) \geq 0$. The roots are $x = 3$ and $x = 4$. Testing intervals around these roots shows that the inequality holds for $x \leq 3$ and $x \geq 4$, hence the correct answer is option C, which represents the union of these intervals.

Question 10

Solve the inequality.
-$x ^ { 2 } - 5 x \leq 0$

Accepted Answer

To solve the inequality $x^2 - 5x \leq 0$, factor out an $x$ to get $x(x - 5) \leq 0$. This implies that the product of $x$ and $x-5$ is non-positive, meaning $x$ is in the interval where it changes sign at the roots $0$ and $5$. By testing values or using a sign chart, we find that the inequality holds for $x$ in the interval $[0, 5]$.

Question 11

Solve the inequality.
-$$x ^ { 2 } + 6 x \leq 0$$

Accepted Answer

To solve the inequality $x^2 + 6x \leq 0$, factor out $x$ to get $x(x + 6) \leq 0$. The roots are $x = 0$ and $x = -6$. The inequality holds between these two roots, including the endpoints, hence the solution is $[ - 6,0 ]$.

Question 12

Solve the problem.
-The quadratic function $$f ( x ) = 0.0040 x ^ { 2 } - 0.43 x + 36.98$$ models the median, or average, age, y, at which U.S. men were first married x years after 1900. In which year was this average age at a minimum? (Round to the nearest
Year.) What was the average age at first marriage for that year? (Round to the nearest tenth.)

Accepted Answer

The minimum value of a quadratic function $f(x) = ax^2 + bx + c$ occurs at $x = -\frac{b}{2a}$. For the given function, $a = 0.0040$ and $b = -0.43$, so the year of minimum average age is $x = -\frac{-0.43}{2(0.0040)} = 53.75$, which rounds to 1954 when added to 1900. To find the minimum average age, substitute $x = 53.75$ back into the function: $f(53.75) = 0.0040(53.75)^2 - 0.43(53.75) + 36.98 \approx 25.4$.

Question 13

Solve the inequality.
-$$x ^ { 2 } + 8 x + 15 > 0$$

Accepted Answer

The answer of Solve the inequality.
-\[x ^ { 2 }...

Question 14

Solve the inequality.
-$$x ^ { 2 } - 25 > 0$$

Accepted Answer

This inequality can be factored as $(x - 5)(x + 5) > 0$, which means $x > 5$ or $x < -5$. Therefore, the solution is $( - \infty , - 5 )$ or $( 5 , \infty )$.

Question 15

Solve the problem.
-If $g ( x ) = x ^ { 2 } - 3 x - 18$, solve $g ( x ) \leq 0$.

Accepted Answer

To solve $g(x) \leq 0$ for $g(x) = x^2 - 3x - 18$, first factor the quadratic equation. Factoring gives $(x + 3)(x - 6) \leq 0$. The roots are $x = -3$ and $x = 6$. The inequality is satisfied between these roots, including the endpoints, hence the solution is $[ - 3,6 ]$.

Question 16

Solve the inequality.
-$30 \left( x ^ { 2 } - 1 \right) > 11 x$

Accepted Answer

First, distribute and move all terms to one side to get a quadratic inequality: $30x^2 - 30 - 11x > 0$, which simplifies to $30x^2 - 11x - 30 > 0$. Factoring or using the quadratic formula gives roots at $x = -\frac{5}{6}$ and $x = \frac{6}{5}$. The inequality is satisfied outside these roots, leading to the intervals $\left( - \infty , - \frac { 5 } { 6 } \right)$ and $\left( \frac { 6 } { 5 } , \infty \right)$.

Question 17

Solve the inequality.
-$$4 x ^ { 2 } - 15 < - 17 x$$

Accepted Answer

Rearrange the inequality to $4x^2 + 17x - 15 < 0$. Solve the quadratic equation $4x^2 + 17x - 15 = 0$ to find the roots $x = -5$ and $x = \frac{3}{4}$. The inequality holds between these roots, so the solution is $(-5, \frac{3}{4})$.

Question 18

Solve the problem.
-If $g ( x ) = 70 x ^ { 2 } - 70$ and $h ( x ) = 51 x$, then solve $g ( x ) > h ( x )$.

Accepted Answer

The answer of Solve the problem.
-If \(g ( x )...

Question 19

Solve the problem.
-The revenue achieved by selling $x$ graphing calculators is figured to be $x ( 28 - 0.2 x )$ dollars. The cost of each calculator is $\$ 20$. How many graphing calculators must be sold to make a profit (revenue - cost) of at least $\$ 75.00 ?$

Accepted Answer

To find the number of calculators $x$ that must be sold to make a profit of at least $75, we first express the profit equation: Profit = Revenue - Cost = $x(28 - 0.2x) - 20x$. Simplifying this gives us the quadratic equation in terms of profit. Setting the profit to be at least $75 and solving for $x$ will give us the range of $x$ that satisfies this condition. The correct range that results from solving this equation is $15 < x < 25$, which corresponds to choice D.

Question 20

Solve the inequality.
-$$x ^ { 2 } - 9 x \geq 0$$

Accepted Answer

To solve the inequality $x^2 - 9x \geq 0$, factor it to get $x(x - 9) \geq 0$. The roots are $x = 0$ and $x = 9$. Testing intervals around these roots, we find that the inequality holds for $x \leq 0$ and $x \geq 9$, leading to the solution $(-\infty, 0]$ or $[9, \infty)$.

Quiz 2: Linear and Quadratic Functions

Solve the problem. -Alan is building a garden shaped like a rectangle with a semicircle attached to one short side. If he has 40 feet of fencing to go around it, what dimensions will give him the maximum area in the garden?

Solve the inequality. - $x ^ { 2 } - 4 x - 5 \leq 0$

Solve the inequality. - $x ^ { 2 } + 4 x \geq 0$

Solve the problem. -If $h ( x ) = x ^ { 2 } + 2 x - 8$ , solve $h ( x ) > 0$ .

Solve the problem. -If $f ( x ) = 6 x ^ { 2 } - 5 x$ and $g ( x ) = 2 x + 3$ , solve $f ( x ) \leq g ( x )$ .

Solve the problem. -If $f ( x ) = 6 x ^ { 2 } - 5 x$ and $g ( x ) = 2 x + 3$ , solve for $f ( x ) = g ( x )$

Solve the inequality. - $x ^ { 2 } - 36 \leq 0$

Solve the inequality. - $64 x ^ { 2 } + 9 < 48 x$

Solve the inequality. - $x ^ { 2 } - 7 x \geq - 12$

Solve the inequality. - $x ^ { 2 } - 5 x \leq 0$

Solve the inequality. - $x ^ { 2 } + 6 x \leq 0$

Solve the inequality. - $x ^ { 2 } + 8 x + 15 > 0$

Solve the inequality. - $x ^ { 2 } - 25 > 0$

Solve the problem. -If $g ( x ) = x ^ { 2 } - 3 x - 18$ , solve $g ( x ) \leq 0$ .

Solve the inequality. - $30 \left( x ^ { 2 } - 1 \right) > 11 x$

Solve the inequality. - $4 x ^ { 2 } - 15 < - 17 x$

Solve the problem. -If $g ( x ) = 70 x ^ { 2 } - 70$ and $h ( x ) = 51 x$ , then solve $g ( x ) > h ( x )$ .

Solve the problem. -The revenue achieved by selling $x$ graphing calculators is figured to be $x ( 28 - 0.2 x )$ dollars. The cost of each calculator is $\$ 20$ . How many graphing calculators must be sold to make a profit (revenue - cost) of at least $\$ 75.00 ?$

Solve the inequality. - $x ^ { 2 } - 9 x \geq 0$

Quiz 2: Linear and Quadratic Functions

Solve the problem. -Alan is building a garden shaped like a rectangle with a semicircle attached to one short side. If he has 40 feet of fencing to go around it, what dimensions will give him the maximum area in the garden?

Solve the inequality. - x2−4x−5≤0x ^ { 2 } - 4 x - 5 \leq 0x2−4x−5≤0

Solve the inequality. - x2+4x≥0x ^ { 2 } + 4 x \geq 0x2+4x≥0

Solve the problem. -If h(x) =x2+2x−8h ( x ) = x ^ { 2 } + 2 x - 8h(x) =x2+2x−8 , solve h(x) >0h ( x ) > 0h(x) >0 .

Solve the problem. -If f(x) =6x2−5xf ( x ) = 6 x ^ { 2 } - 5 xf(x) =6x2−5x and g(x) =2x+3g ( x ) = 2 x + 3g(x) =2x+3 , solve f(x) ≤g(x) f ( x ) \leq g ( x ) f(x) ≤g(x) .

Solve the problem. -If f(x) =6x2−5xf ( x ) = 6 x ^ { 2 } - 5 xf(x) =6x2−5x and g(x) =2x+3g ( x ) = 2 x + 3g(x) =2x+3 , solve for f(x) =g(x) f ( x ) = g ( x ) f(x) =g(x)

Solve the inequality. - x2−36≤0x ^ { 2 } - 36 \leq 0x2−36≤0

Solve the inequality. - 64x2+9<48x64 x ^ { 2 } + 9 < 48 x64x2+9<48x

Solve the inequality. - x2−7x≥−12x ^ { 2 } - 7 x \geq - 12x2−7x≥−12

Solve the inequality. - x2−5x≤0x ^ { 2 } - 5 x \leq 0x2−5x≤0

Solve the inequality. - x2+6x≤0x ^ { 2 } + 6 x \leq 0x2+6x≤0

Solve the inequality. - x2+8x+15>0x ^ { 2 } + 8 x + 15 > 0x2+8x+15>0

Solve the inequality. - x2−25>0x ^ { 2 } - 25 > 0x2−25>0

Solve the problem. -If g(x) =x2−3x−18g ( x ) = x ^ { 2 } - 3 x - 18g(x) =x2−3x−18 , solve g(x) ≤0g ( x ) \leq 0g(x) ≤0 .

Solve the inequality. - 30(x2−1) >11x30 \left( x ^ { 2 } - 1 \right) > 11 x30(x2−1) >11x

Solve the inequality. - 4x2−15<−17x4 x ^ { 2 } - 15 < - 17 x4x2−15<−17x

Solve the problem. -If g(x) =70x2−70g ( x ) = 70 x ^ { 2 } - 70g(x) =70x2−70 and h(x) =51xh ( x ) = 51 xh(x) =51x , then solve g(x) >h(x) g ( x ) > h ( x ) g(x) >h(x) .

Solve the inequality. - x2−9x≥0x ^ { 2 } - 9 x \geq 0x2−9x≥0