[scheduled:: 2025-01-14]
Word Problems with Inequalities
Links and useful resources
- gr7's Prealgebra Course Outline START HERE<<<
- IXL Grade 7 index
- IXL Grade 8 index
- AoPS Alcumus
- AoPS Prealgebra Outline with assignments
- Handwritten math converter
- IXL prealgebra practice index
- OpenSTAX Prealgebra-1 textbook
Lesson-specific resource links
- Solving systems of inequalities
- Khan practice with systems of inequalities
- IXL Lesson: Systems of linear inequalities
Concept summary and connections
- linear inequalities
- system of linear inequalities
- solving inequality problems
Solving a system of inequalities means that you find out what the shape is of a chunk of space that holds all of the points that satisfy all of the inequalities. It's not a case where you'll find a single number (usually). That means that many of the tricks we use from equations in order to get to a single solution don't really apply here (you can't "solve for y and substitute it into the other equation" because there isn't a clear way to combine them).
- What we do instead is find the lines that make the border of the area that's okay, and find the points that make the corners.
- There will be one line per equation
- You can have more equations than variables with inequalities, and it's fine. You can also have less equations than variables - all we're doing is cutting off chunks of space that don't fit and leaving the rest!
- If the inequality is
or (strictly greater or less), draw the line as a dashed line to show that values on the line aren't included. - If the inequality is
or (greater or equal, or less or equal), draw the line as solid because values on the line are included. - Each time you add a line, test a point on one side of it to see which side of the line is allowed.
- Once you figure out which side of the line satisfies that inequality, shade it a bit. If there are already other regions shaded, just add more shading to the parts that overlap.
- Once all of the lines have been added, you can mark all of the points where they intersect and darken the lines between (make sure to keep the dashed ones dashed!)
- That is your answer - it'll be something between an infinite wedge shape and a multi-sided polygon.
Worked examples
Graph these inequalities and figure out what the valid region is:
Media resources
- Youtube search for "linear inequalities"
- Youtube search for "system of linear inequalities"
- Youtube search for "solving inequality problems"