Question

solve and graph the following linear inequalities for the unknown variable. note: all necessary algebraic operations required to isolate the variable must be shown to receive full credit. (a) $z - 9 \leq 23$ (10 points) (b) $\frac{7 - 4y}{2} > 5$ (10 points) (c) $6x + 42 \geq 66$ (10 points) (d) $14x - 32y + 57z \geq 65$ for $x = 7$ and $z = 3$ (10 points)

Explanation:

Part (a)

Step 1: Isolate \( z \)

To solve \( z - 9 \leq 23 \), we add 9 to both sides of the inequality.
\( z - 9 + 9 \leq 23 + 9 \)

Step 2: Simplify

Simplifying both sides gives \( z \leq 32 \).

Step 1: Eliminate the denominator

Multiply both sides of \( \frac{7 - 4y}{2} > 5 \) by 2 (since 2 is positive, the inequality sign remains the same).
\( 2\times\frac{7 - 4y}{2} > 5\times2 \)
\( 7 - 4y > 10 \)

Step 2: Isolate the term with \( y \)

Subtract 7 from both sides:
\( 7 - 4y - 7 > 10 - 7 \)
\( -4y > 3 \)

Step 3: Solve for \( y \)

Divide both sides by -4. When dividing by a negative number, the inequality sign flips.
\( \frac{-4y}{-4} < \frac{3}{-4} \)
\( y < -\frac{3}{4} \)

Step 1: Isolate the term with \( x \)

Subtract 42 from both sides of \( 6x + 42 \geq 66 \):
\( 6x + 42 - 42 \geq 66 - 42 \)
\( 6x \geq 24 \)

Step 2: Solve for \( x \)

Divide both sides by 6:
\( \frac{6x}{6} \geq \frac{24}{6} \)
\( x \geq 4 \)

Answer:

\( z \leq 32 \) (To graph, draw a number line, place a closed circle at 32, and shade to the left.)

Part (b)