Problem 1: Test a point
Determine whether \((2, 1)\) is a solution of \(y<3x\).
Problem 2: Points on the boundary
Determine whether each point is a solution of \(2x+3y\ge 12\).
- \((3, 2)\)
- \((1, 2)\)
Problem 3: Solid or dashed
State whether the boundary line is solid or dashed for each inequality.
- \(y>2x+1\)
- \(y\le -x+4\)
- \(y<\frac{1}{2}x\)
- \(y\ge 3x-2\)
Problem 4: Graph y > x + 2
Graph \(y>x+2\).
Problem 5: Graph y ≤ −2x + 3
Graph \(y\le -2x+3\).
Problem 6: Graph with a fractional slope
Graph \(y\ge \frac{1}{2}x-1\).
Problem 7: Standard form
Graph \(2x+3y<6\) by finding the intercepts of the boundary and testing a point.
Problem 8: Divide by a negative
Rewrite \(-3y\ge 6x-9\) in the form \(y\le mx+b\) or \(y\ge mx+b\). Then state whether the line is solid or dashed and which side is shaded.
Problem 9: Vertical and horizontal
Graph \(x\ge -2\) on one grid and \(y<3\) on another grid.
Problem 10: Write from a graph: dashed
Write the inequality shown by the graph.
Problem 11: Write from a graph: solid
Write the inequality shown by the graph.
Problem 12: Check points in a system
Which of the points \((0, 0)\), \((3, 1)\), \((2, 2)\), \((4, 2)\) and \((-1, 4)\) are solutions of the system \(y>x-1\) and \(y\le -x+5\)?
Problem 13: Graph a system
Graph the system \(y<-x+4\) and \(y>2x-5\). Label the point where the boundaries meet and tell whether it is a solution.
Problem 14: A triangular region
Graph the system \(x\ge 0\), \(y\ge 0\) and \(x+y\le 6\). Name the corners of the region and tell whether \((2, 3)\) is a solution.
Problem 15: A rectangular region
Graph \(-2\le x\le 3\) and \(1\le y\le 4\). List the corners and find the area of the region.
Problem 16: No overlap
Explain why the system \(y>x+3\) and \(y<x-1\) has no solution.
Problem 17: Concert budget
Balcony tickets cost \(\$15\) and floor tickets cost \(\$25\). A club has at most \(\$150\) to spend. Let \(x\) be balcony tickets and \(y\) floor tickets. Write an inequality, then decide whether buying (a) 4 balcony and 3 floor, (b) 5 balcony and 3 floor, (c) 2 balcony and 5 floor tickets stays within the budget.
Problem 18: Bake sale limits
A bake sale can make at most 60 items. Let \(x\) be muffins and \(y\) be cookies. At least 20 cookies are needed. Write a system of inequalities, then decide which of \((35, 25)\), \((45, 20)\), \((30, 15)\) and \((20, 20)\) are possible.
Problem 19: Yard work hours
Diego has at most 10 hours this week. A lawn takes 2 hours and a dog walk takes 1 hour. Let \(x\) be lawns and \(y\) be dog walks. Write a system, name the corners of the feasible region, and decide whether 3 lawns with 4 walks, or 4 lawns with 3 walks, fits his time.
Problem 20: Lemonade goal
Aisha sells cups of lemonade for \(\$2\) and cookies for \(\$3\). She wants to earn at least \(\$60\) and can bake at most 25 cookies. Let \(x\) be cups and \(y\) be cookies. Write a system and decide whether selling (15, 10), (10, 10) or (30, 25) meets both conditions.
Problem 21: Find the error
Liam graphs \(3x-2y>6\) with a solid line and shades the side that contains the origin. Describe both mistakes, then graph the inequality correctly.
Problem 22: Garden fence
Noah has at most 40 feet of fence for a rectangular garden. The length \(x\) must be at least 8 feet and the width \(y\) at least 4 feet. Write a system of inequalities, find the corners of the feasible region, and check whether a 10 ft by 8 ft garden is allowed.
See every chapter on the Algebra 1 math page.