Unit 6: Inequalities
An inequality keeps a whole range of numbers true at once, drawn as a ray or a shaded region instead of a single point.
6 topics · 22 questions
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Solving One-Variable Inequalities
- How it worksHow does isolating x in an inequality work the same way it does for an equation?
- Worked exampleSolve the inequality 3x - 5 is greater than 10 and graph the solution on a number line.
- Common mistakeWhy doesn't an inequality's solution stop being true once you plug in a really large number?
- Going furtherWhat happens to the solution set when both sides of an inequality are multiplied by a positive fraction?
Your turn1 of 2
Solve the inequality 7x - 6 is greater than 28. Find the boundary value of x (where equality holds).Why You Flip the Inequality Sign
- How it worksHow does multiplying an inequality by a negative number reverse the order of every number on the line?
- Worked exampleSolve the inequality -2x + 3 is less than 11, flipping the sign at the right step.
- Common mistakeWhy doesn't adding a negative number to both sides of an inequality require flipping the sign?
- Going furtherWhat happens to a compound inequality's sign direction when every part gets divided by a negative number?
Your turn1 of 2
Solve the inequality -7x + 10 is less than 19. Find the boundary value of x.Compound Inequalities
- How it worksHow does an \"and\" compound inequality require both conditions to be true at once on the number line?
- Worked exampleSolve the compound inequality -3 is less than 2x + 1 and 2x + 1 is less than 9.
- How it worksHow does an \"or\" compound inequality end up covering two separate rays on the number line?
- Common mistakeWhy isn't every compound inequality's graph one single connected piece of the number line?
Your turn1 of 2
Solve the compound inequality -3 is less than 4x + 8 and 4x + 8 is less than 19. Find the upper boundary value of x.Graphing Two-Variable Inequalities
- How it worksHow does testing a point decide which side of the boundary line to shade?
- Worked exampleGraph the inequality y is greater than 2x - 1, including whether the boundary line is solid or dashed.
- Common mistakeWhy does a strict inequality need a dashed boundary line instead of a solid one?
- Going furtherWhat happens to a shaded region when the inequality symbol switches from greater than to greater than or equal to?
Your turn1 of 2
To graph y > 7x + -4 (with -4 negative), you test the point (0, 0). Substituting gives 0 > -4, which is true. What do you conclude?Systems of Two-Variable Inequalities
- How it worksHow do you find the overlapping shaded region for a system of two inequalities?
- Worked exampleGraph the solution to the system y is less than x + 3 and y is greater than or equal to -x + 1.
- Going furtherWhat happens to a system of inequalities' solution region when one boundary line becomes parallel to the other?
Your turn1 of 2
A point satisfies inequality 1 of a system but fails inequality 2. Where does that put it relative to the system's solution region?Inequality Word Problems
- How it worksHow do you decide whether a word problem's limit should translate to greater-than or less-than-or-equal-to?
- Worked exampleFind how many rides a 40 dollar budget allows at a fair charging 5 dollars entry plus 2.50 dollars per ride.
- Going furtherWhat happens to the inequality's solution set if the entry fee in the fair problem doubles?
Your turn1 of 2
A budget of 68 dollars allows entry for 8 dollars plus 5 dollars per ride. Find the value of r (rides) where the entry fee plus ride cost together exactly use up the budget.