Question

1. solve the inequality. graph its solution.

-2|n - 4| > -2

1. evaluate the function.

h(x)=4x + 2; find h(-5)

1. solve the system:

3x - 16y=-23
-2x + 8y = 10

solve the system:
-2a + 3b - 5c=-9
3a + 6b + 5c = 30
-2b + c = 7

Explanation:

1. For the inequality \(-2|n - 4|>-2\):

• Step 1: Divide by \(-2\) and reverse the sign
• Divide both sides by \(-2\): \(\frac{-2|n - 4|}{-2}<\frac{-2}{-2}\), resulting in \(|n - 4|<1\).
• Step 2: Rewrite as a compound - inequality
• \(|n - 4|<1\) is equivalent to \(-1 < n - 4<1\).
• Step 3: Solve the compound - inequality
• Add \(4\) to all parts: \(n\in(3,5)\).

1. For the function \(h(x)=4x + 2\), find \(h(-5)\):

• Step 1: Substitute \(x=-5\)
• \(h(-5)=4\times(-5)+2\).
• Step 2: Perform multiplication
• \(4\times(-5)=-20\).
• Step 3: Perform addition
• \(-20 + 2=-18\).

1. **For the system of equations \(

$$\begin{cases}3x-16y=-23\\-2x + 8y=10\end{cases}$$

\)**:

• Step 1: Make \(y\) coefficients opposite
• Multiply the second equation by \(2\) to get \(-4x+16y = 20\).
• Step 2: Add the equations
• \((3x-16y)+(-4x + 16y)=-23 + 20\).
• Step 3: Solve for \(x\)
• \(-x=-3\), so \(x = 3\).
• Step 4: Substitute \(x\) into an equation
• Substitute \(x = 3\) into \(-2x + 8y=10\).
• Step 5: Solve for \(y\)
• \(-6 + 8y=10\), then \(y = 2\).

1. **For the system of equations \(

$$\begin{cases}-2a+3b-5c=-9\\3a + 6b+5c=30\\-2b + c=7\end{cases}$$

\)**:

• Step 1: Eliminate \(c\) from the first two equations
• Add the first and second equations: \((-2a+3b-5c)+(3a + 6b+5c)=-9 + 30\).
• Step 2: Express \(a\) in terms of \(b\)
• \(a=21-9b\).
• Step 3: Express \(c\) in terms of \(b\)
• From \(-2b + c=7\), we have \(c=7 + 2b\).
• Step 4: Substitute \(a\) and \(c\) into the first equation
• \(-2(21-9b)+3b-5(7 + 2b)=-9\).
• Step 5: Expand and simplify
• Expand and combine like - terms to solve for \(b\), then find \(a\) and \(c\).

Answer:

1. For the inequality \(-2|n - 4|>-2\):

• First, divide both sides of the inequality by \(- 2\). Remember that when dividing an inequality by a negative number, the direction of the inequality sign changes. So we get \(|n - 4|<1\).
• This is equivalent to \(-1 < n - 4<1\).
• Add \(4\) to all parts of the compound - inequality: \(-1+4 < n-4 + 4<1 + 4\), which simplifies to \(3 < n<5\).

1. For the function \(h(x)=4x + 2\), find \(h(-5)\):

• Substitute \(x=-5\) into the function \(h(x)\).
• \(h(-5)=4\times(-5)+2\).
• First, calculate \(4\times(-5)=-20\).
• Then, \(-20 + 2=-18\).

1. **For the system of equations \(

$$\begin{cases}3x-16y=-23\\-2x + 8y=10\end{cases}$$

\)**:

• Multiply the second equation by \(2\): \(2(-2x + 8y)=2\times10\), which gives \(-4x+16y = 20\).
• Add this new equation to the first equation: \((3x-16y)+(-4x + 16y)=-23 + 20\).
• Combine like - terms: \(3x-4x-16y + 16y=-3\), so \(-x=-3\), and \(x = 3\).
• Substitute \(x = 3\) into the second equation \(-2\times3+8y=10\).
• Calculate \(-6 + 8y=10\).
• Add \(6\) to both sides: \(8y=10 + 6=16\).
• Divide both sides by \(8\): \(y = 2\).

1. **For the system of equations \(

$$\begin{cases}-2a+3b-5c=-9\\3a + 6b+5c=30\\-2b + c=7\end{cases}$$

\)**:

• Add the first and second equations: \((-2a+3b-5c)+(3a + 6b+5c)=-9 + 30\).
• Combine like - terms: \(-2a+3a+3b+6b-5c + 5c=21\), so \(a + 9b=21\), which gives \(a=21-9b\).
• From the third equation \(c=7 + 2b\).
• Substitute \(a = 21-9b\) and \(c=7 + 2b\) into the first equation: \(-2(21-9b)+3b-5(7 + 2b)=-9\).
• Expand: \(-42 + 18b+3b-35-10b=-9\).
• Combine like - terms: \(18b+3b-10b=-9 + 42+35\).
• \(11b=68\), so \(b=\frac{68}{11}\).
• Then \(a=21-9\times\frac{68}{11}=\frac{231-612}{11}=-\frac{381}{11}\).
• And \(c=7+2\times\frac{68}{11}=\frac{77 + 136}{11}=\frac{213}{11}\).