Special Cases: No Solution and All Real Numbers in Inequalities
Grade 9 students in California Reveal Math Algebra 1 learn to recognize two special cases when solving multi-step inequalities: no solution and all real numbers. When solving, if the variable terms cancel and a false statement remains like 6≥9, the inequality has no solution (empty set). If a true statement remains like -3<2, every real number is a solution. For example, 2(x+3)≥2x+9 simplifies to 6≥9 — false, so no solution; while 3(x-1)<3x+2 simplifies to -3<2 — true, so all real numbers satisfy the inequality.
Key Concepts
Property When solving a multi step inequality, sometimes the variable terms completely cancel each other out. When this happens, look at the remaining numerical statement: Contradiction (False Statement): If the statement is mathematically false (e.g., $5 < 2$), there is No Solution ($\emptyset$). Identity (True Statement): If the statement is mathematically true (e.g., $4 \geq 4$ or $ 1 < 5$), the solution is All Real Numbers $( \infty, \infty)$.
Examples Example 1 (No Solution): Solve $2(x + 3) \geq 2x + 9$. Distribute: $2x + 6 \geq 2x + 9$. Subtract $2x$ from both sides: $6 \geq 9$. Since 6 is not greater than or equal to 9, this is a false statement. The solution set is $\emptyset$ (No Solution). Example 2 (All Real Numbers): Solve $3(x 1) < 3x + 2$. Distribute: $3x 3 < 3x + 2$. Subtract $3x$ from both sides: $ 3 < 2$. Since 3 is always less than 2, this is a true statement. The solution is All Real Numbers.
Explanation Don't panic if your variable completely disappears while solving! When this happens, the math is telling you that the specific value of the variable doesn't actually matter. Your only job is to judge what is left. If the leftover numbers make a true statement, then literally any real number will work. If they make a false statement, then no number in the world can fix it.
Common Questions
What are the two special cases when solving inequalities?
If all variables cancel and a false statement remains (like 5<2), the solution set is empty — no solution. If a true statement remains (like -3<2 or 4≥4), every real number satisfies the inequality.
How do you recognize no solution in an inequality?
Solve the inequality until the variable terms cancel. If what remains is a false numerical statement, such as 6≥9, there is no solution and the solution set is the empty set.
How do you recognize all real numbers as the solution?
Solve the inequality until the variable terms cancel. If what remains is a true numerical statement, such as -3<2, then all real numbers satisfy the inequality.
How do you solve 2(x+3)≥2x+9?
Distribute: 2x+6≥2x+9. Subtract 2x: 6≥9. This is false, so there is no solution.
How do you solve 3(x-1)<3x+2?
Distribute: 3x-3<3x+2. Subtract 3x: -3<2. This is always true, so the solution is all real numbers.
Which unit covers these special cases in Algebra 1?
This skill is from Unit 5: Linear Inequalities in California Reveal Math Algebra 1, Grade 9.