Question

tsia linear questions (sept 11)

1. solve the inequality and graph the solution.

( 3(y - 4) leq 6 )
plot the endpoints. select an endpoint to change it from closed to open. select the middle of the segment, ray, or line to delete it.
← -10 -8 -6 -4 -2 0 2 4 6 8 10 →

1. solve using substitution.

( y = x - 3 )
( y = 2x - 8 )
( (square, square) )

rewrite the following equation in slope-intercept form.
( y - 6 = 8(x - 5) )
write your answer using integers, proper fractions, and improper fractions in simplest form.

Explanation:

Problem 1: Solve the inequality \( 3(y - 4) \leq 6 \) and graph the solution.

Step 1: Divide both sides by 3

To isolate the term with \( y \), divide both sides of the inequality by 3.
\[
\frac{3(y - 4)}{3} \leq \frac{6}{3}
\]
Simplifying, we get:
\[
y - 4 \leq 2
\]

Step 2: Add 4 to both sides

To solve for \( y \), add 4 to both sides of the inequality.
\[
y - 4 + 4 \leq 2 + 4
\]
Simplifying, we get:
\[
y \leq 6
\]

Step 1: Substitute \( y = x - 3 \) into the second equation

Since both expressions equal \( y \), we can set them equal to each other:
\[
x - 3 = 2x - 8
\]

Step 2: Solve for \( x \)

Subtract \( x \) from both sides:
\[
-3 = x - 8
\]
Add 8 to both sides:
\[
x = 5
\]

Step 3: Substitute \( x = 5 \) back into \( y = x - 3 \)

\[
y = 5 - 3
\]
Simplifying, we get:
\[
y = 2
\]

Step 1: Distribute the 8 on the right - hand side

Using the distributive property \( a(b - c)=ab - ac \), we have:
\[
y - 6 = 8x - 40
\]

Step 2: Add 6 to both sides

To solve for \( y \), add 6 to both sides of the equation:
\[
y=8x - 40 + 6
\]
Simplifying the right - hand side:
\[
y = 8x-34
\]

Answer:

The solution to the inequality is \( y \leq 6 \). To graph this, we plot a closed circle at \( y = 6 \) (since the inequality is "less than or equal to") and draw a ray to the left of 6 on the number line.

Problem 2: Solve the system of equations using substitution: \(

$$\begin{cases} y = x - 3 \\ y = 2x - 8 \end{cases}$$

\)