Question

y > 4x + 6. for which of the following tables are all the values of x and their corresponding values of y solutions to the given inequality? a) x: 2, y: 19; x: 4, y: 30; x: 6, y: 41 b) x: 2, y: 8; x: 4, y: 16; x: 6, y: 24 c) x: 2, y: 13; x: 4, y: 18; x: 6, y: 23 d) x: 2, y: 13; x: 4, y: 21; x: 6, y: 29. the function h is defined by h(x)=\frac{8}{5x + 6}. what is the value of h(2)? the figure shows the lengths, in inches, of two sides of a right - triangle. what is the area of the triangle, in square inches? note: figure not drawn to scale. which expression is equivalent to (x² + 11)²+(x - 5)(x + 5)? a) x⁴+23x² - 14 b) x⁴+23x² + 96 c) x⁴+12x² + 121 d) x⁴+x² + 146

Explanation:

Step1: Evaluate the inequality \(y > 4x+6\) for each table - Option A

For \(x = 2\), \(4x+6=4\times2 + 6=8 + 6=14\), and \(y = 19\), \(19>14\). For \(x = 4\), \(4x+6=4\times4+6=16 + 6=22\), and \(y = 20\), \(20<22\) (so option A is not correct).

Step2: Evaluate the inequality for Option B

For \(x = 2\), \(4x+6=14\), and \(y = 8\), \(8<14\) (so option B is not correct).

Step3: Evaluate the inequality for Option C

For \(x = 2\), \(4x+6=14\), and \(y = 13\), \(13<14\) (so option C is not correct).

Step4: Evaluate the inequality for Option D

For \(x = 2\), \(4x+6=14\), and \(y = 15\), \(15>14\). For \(x = 4\), \(4x+6=22\), and \(y = 21\), \(21>22\). For \(x = 6\), \(4x+6=4\times6+6=24 + 6=30\), and \(y = 29\), \(29>30\). So the table in option D has all values of \(x\) and \(y\) as solutions to \(y>4x + 6\).

Step1: Find \(h(2)\) for the function \(h(x)=\frac{8}{5x+6}\)

Substitute \(x = 2\) into the function \(h(x)\). We get \(h(2)=\frac{8}{5\times2+6}\).

Step2: Simplify the denominator

Calculate \(5\times2+6=10 + 6=16\).

Step3: Find the value of \(h(2)\)

\(h(2)=\frac{8}{16}=\frac{1}{2}\)

Step1: Expand \((x^{2}+11)^{2}\)

Using the formula \((a + b)^{2}=a^{2}+2ab + b^{2}\), where \(a=x^{2}\) and \(b = 11\), we have \((x^{2}+11)^{2}=(x^{2})^{2}+2\times x^{2}\times11+11^{2}=x^{4}+22x^{2}+121\).

Step2: Expand \((x - 5)(x + 5)\)

Using the difference - of - squares formula \((a - b)(a + b)=a^{2}-b^{2}\), where \(a=x\) and \(b = 5\), we get \((x - 5)(x + 5)=x^{2}-25\).

Step3: Combine the two expanded expressions

\((x^{2}+11)^{2}+(x - 5)(x + 5)=x^{4}+22x^{2}+121+x^{2}-25=x^{4}+23x^{2}+96\)

Step1: Recall the area formula for a right - triangle

The area formula for a right - triangle is \(A=\frac{1}{2}ab\), where \(a\) and \(b\) are the lengths of the two legs of the right - triangle.

Step2: Identify the legs of the right - triangle

The lengths of the two legs are \(a = 3\) and \(b = 5\).

Step3: Calculate the area

\(A=\frac{1}{2}\times3\times5=\frac{15}{2}=7.5\)

Answer:

D