➕ Algebra 1 · Inequalities
Compound Inequalities flashcards
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All 16 flashcards at a glance
Card 1
What does “and” mean in a compound inequality, and what does its graph look like?
Both inequalities must be true at the same time. The graph keeps only the overlap, the part shaded by both pieces.
Card 2
What does “or” mean in a compound inequality, and what does its graph look like?
At least one inequality must be true (both is allowed). The graph keeps everything shaded by either piece.
Card 3
How do you read 12 ≤ x < 30?
“x is at least 12 and less than 30.” It means x ≥ 12 and x < 30: a closed circle at 12, an open circle at 30, and shading between.
Card 4
Solve 6 ≤ x + 2 ≤ 15.
Subtract 2 from all three parts: 4 ≤ x ≤ 13.
Card 5
Solve 9 ≤ 3x < 21.
Divide all three parts by 3: 3 ≤ x < 7.
Card 6
Solve −6 ≤ −2x ≤ 14.
Divide all three parts by −2 and reverse both signs: 3 ≥ x ≥ −7. Written with the smaller number first: −7 ≤ x ≤ 3.
Card 7
Write “x is less than 2 or greater than 9” with symbols. Is 5 a solution?
x < 2 or x > 9. No: 5 < 2 is false and 5 > 9 is false.
Card 8
Can you write x < 2 or x > 9 as 9 < x < 2?
No. 9 < x < 2 means x > 9 and x < 2, which no number satisfies. An “or” with two rays pointing away from each other cannot be written as one three-part inequality.
Card 9
What does it mean when the pieces of an “and” do not overlap, like x < 0 and x > 6?
No number satisfies both, so there is no solution.
Card 10
What are the solutions of x ≤ 12 or x ≥ 4?
Every real number. A number that is not at most 12 is greater than 12, so it is at least 4.
Card 11
What are the solutions of x ≥ 11 and x ≤ 11? And of x < 11 or x > 11?
The first: only x = 11. The second: every real number except 11.
Card 12
What is the solution of x > 6 and x > 10? And of x > 6 or x > 10?
“And”: x > 10, because a number must satisfy both. “Or”: x > 6, because any number greater than 10 is already greater than 6.
Card 13
Write 1 ≤ x < 10 in interval notation.
[1, 10). The square bracket keeps 1, and the parenthesis leaves out 10.
Card 14
Write x ≤ 0 or x > 8 in interval notation.
(−∞, 0] ∪ (8, ∞). The symbol ∪ means “or” (union), and infinity always gets a parenthesis.
Card 15
How can you check an answer like −1 ≤ x < 8?
Test a number inside (like 0), which should pass every piece. Test numbers outside (like −2 and 9), which should each fail a piece. Test the boundaries: −1 passes because of ≤, and 8 fails because of <.
Card 16
In this lesson’s SPLIT memory trick, what do the letters stand for?
Split a three-part inequality and solve each piece. Plot each piece on its own number line. Look at the joining word. Intersect for “and” (keep the overlap). Take everything shaded for “or”.