Question

graph each inequality in the coordinate plane.

1. ( x leq 4 ) 23. ( y geq -1 ) 24. ( x > -2 ) 25. ( y < -4 )
2. ( -2x + y geq 3 ) 27. ( x + 3y < 15 ) 28. ( 4x - y > 2 ) 29. ( -x + 0.25y leq -1.75 )
3. carpentry you budget $200 for wooden planks for outdoor furniture. cedar costs $2.50 per foot and pine costs $1.75 per foot. let ( x ) = the number of feet of cedar and let ( y ) = the number of feet of pine. what is an inequality that shows how much of each type of wood can be bought? graph the inequality. what are three possible amounts of each type of wood that can be bought within your budget?
4. business a fish market charges $9 per pound for cod and $12 per pound for flounder. let ( x ) = the number of pounds of cod. let ( y ) = the number of pounds of flounder. what is an inequality that shows how much of each type of fish the store must sell today to reach a daily quota of at least $120? graph the inequality. what are three possible amounts of each fish that would satisfy the quota?

write a linear inequality that represents each graph.

1. image of a graph 33. image of a graph 34. image of a graph
2. think about a plan a truck that can carry no more than 6400 lb is being used to...

Explanation:

For problems 22-29 (Graphing inequalities):

Step1: Identify boundary line

For $x \leq 4$: Boundary is $x=4$ (vertical line, solid since $\leq$).
For $y \geq -1$: Boundary is $y=-1$ (horizontal line, solid since $\geq$).
For $x > -2$: Boundary is $x=-2$ (vertical line, dashed since $>$).
For $y < -4$: Boundary is $y=-4$ (horizontal line, dashed since $<$).
For $-2x + y \geq 3$: Rewrite to $y \geq 2x + 3$, boundary $y=2x+3$ (solid).
For $x + 3y < 15$: Rewrite to $y < -\frac{1}{3}x + 5$, boundary $y=-\frac{1}{3}x+5$ (dashed).
For $4x - y > 2$: Rewrite to $y < 4x - 2$, boundary $y=4x-2$ (dashed).
For $-x + 0.25y \leq -1.75$: Rewrite to $y \leq 4x - 7$, boundary $y=4x-7$ (solid).

Step2: Test a point for shading

For $x \leq 4$: Test $(0,0)$: $0 \leq 4$ is true, shade left of $x=4$.
For $y \geq -1$: Test $(0,0)$: $0 \geq -1$ is true, shade above $y=-1$.
For $x > -2$: Test $(0,0)$: $0 > -2$ is true, shade right of $x=-2$.
For $y < -4$: Test $(0,0)$: $0 < -4$ is false, shade below $y=-4$.
For $y \geq 2x + 3$: Test $(0,0)$: $0 \geq 3$ is false, shade above the line.
For $y < -\frac{1}{3}x + 5$: Test $(0,0)$: $0 < 5$ is true, shade below the line.
For $y < 4x - 2$: Test $(0,0)$: $0 < -2$ is false, shade below the line.
For $y \leq 4x - 7$: Test $(0,0)$: $0 \leq -7$ is false, shade below the line.

Step1: Write cost inequality

Total cost = (cedar cost) + (pine cost) $\leq$ budget
$\$2.50x + \$1.75y \leq \$200$
Simplify: $10x + 7y \leq 800$ (multiply all terms by 4)

Step2: Graph the inequality

Boundary line: $10x + 7y = 800$ (solid, since $\leq$). Intercepts: $x=80$ (when $y=0$), $y=\frac{800}{7}\approx114.29$ (when $x=0$). Test $(0,0)$: $0 \leq 800$ is true, shade below the line (only positive $x,y$ since length can't be negative).

Step3: Find possible amounts

Choose non-negative $(x,y)$ that satisfy the inequality:

1. $x=0, y=114$: $2.50(0)+1.75(114)=199.5 \leq 200$
2. $x=40, y=57$: $2.50(40)+1.75(57)=100+99.75=199.75 \leq 200$
3. $x=80, y=0$: $2.50(80)+1.75(0)=200 \leq 200$

Step1: Write revenue inequality

Total revenue = (cod revenue) + (flounder revenue) $\geq$ quota
$\$9x + \$12y \geq \$120$
Simplify: $3x + 4y \geq 40$ (divide all terms by 3)

Step2: Graph the inequality

Boundary line: $3x + 4y = 40$ (solid, since $\geq$). Intercepts: $x=\frac{40}{3}\approx13.33$ (when $y=0$), $y=10$ (when $x=0$). Test $(0,0)$: $0 \geq 40$ is false, shade above the line (only positive $x,y$ since weight can't be negative).

Step3: Find possible amounts

Choose non-negative $(x,y)$ that satisfy the inequality:

1. $x=0, y=10$: $9(0)+12(10)=120 \geq 120$
2. $x=10, y=3$: $9(10)+12(3)=90+36=126 \geq 120$
3. $x=20, y=0$: $9(20)+12(0)=180 \geq 120$

Answer:

Graphs are constructed as described:

• 22: Solid vertical line $x=4$, shade left.
• 23: Solid horizontal line $y=-1$, shade above.
• 24: Dashed vertical line $x=-2$, shade right.
• 25: Dashed horizontal line $y=-4$, shade below.
• 26: Solid line $y=2x+3$, shade above.
• 27: Dashed line $y=-\frac{1}{3}x+5$, shade below.
• 28: Dashed line $y=4x-2$, shade below.
• 29: Solid line $y=4x-7$, shade below.

---

Problem 30 (Carpentry Budget)