Equations and inequalities
First-degree equations and inequalities, zero product, quadratics with the discriminant, quotients and sign tables, and writing the solution set as intervals.
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First-degree equation: what is it?
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An equation that reduces to ax + b = 0 with a not zero, and then has a single solution, x = −b/a.
Isolate x by doing the same operation on both sides: 3x + 6 = 0 gives 3x = −6 then x = −2.
What are the solutions of (x − 3)(2x + 4) = 0?
- 3 and −2
- 3 and 2
- −3 and −2
- −3 and 2
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3 and −2
x − 3 = 0 gives x = 3 and 2x + 4 = 0 gives 2x = −4, so x = −2. Copying the −3 from the bracket without flipping its sign gives −3, dropping the sign when moving 4 across gives 2.
A product is zero if and only if … is zero: (x + 5)(3x − 6) = 0 gives x = −5 or x = 2.
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A product is zero if and only if one of its factors is zero: (x + 5)(3x − 6) = 0 gives x = −5 or x = 2.
This is why you factor before solving. Never expand an equation that is already a product equal to zero.
Quadratic equation: what is it?
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An equation ax² + bx + c = 0 with a not zero. It has at most two real solutions, which the discriminant lets you find.
When Δ > 0 the solutions are (−b − √Δ) / 2a and (−b + √Δ) / 2a. Setting a product to zero is quicker when the expression is already factored.
What is the solution set of −2x + 6 > 0?
- (3, +∞)
- (−∞, 3)
- (−∞, −3)
- (−3, +∞)
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(−∞, 3)
−2x > −6, then divide by −2 and reverse the direction: x < 3. Forgetting the reversal gives (3, +∞).