1.02D - Square Root Inequalities | Lecture Note

University:
High School
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High School Physics
Academic year:

2024
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1.02D: INEQUALITIES Square Root Function • A visual is very helpful Ex: 1 (1) Visual −2 𝑥 − 3 + 4 ≤ 2 R: 𝑥 ≥ 3 (2) Restrictions (3) Solve Process −2 𝑥 − 3 ≤ −2 𝑥−3≥1 𝑥−3≥1 𝑥≥4 𝑓 𝑥 = −2 𝑥 − 3 + 4 𝑔 𝑥 =2 𝑓 𝑥 ≤ 𝑔(𝑥) Ex 1 What if Greater than? • A visual is very helpful Ex: 1 - revisited (1) Visual −2 𝑥 − 3 + 4 ≥ 2 R: 𝑥 ≥ 3 (2) Restrictions (3) Solve −2 𝑥 − 3 ≥ −2 𝑥−3≤1 𝑥−3≤1 𝑥≤4 2 𝑥+4−1>𝑥 𝑓 𝑥 =2 𝑥+4−1 𝑔 𝑥 =𝑥 𝑓 𝑥 > 𝑔(𝑥) Ex 2 • A visual is very helpful (1) Visual (2) Restrictions (3) Solve (and validate) Ex: 2 2 𝑥+4−1>𝑥 R: 𝑥 ≥ −4 2 𝑥+4>𝑥+1 𝑥+1 𝑥+4> 2 𝑥+1 Restriction: 2 ≥ 0 𝑜𝑟 𝑥 ≥ −1 𝑥 2 + 2𝑥 + 1 𝑥+4> 4 4𝑥 + 16 > 𝑥 2 + 2𝑥 + 1 0 > 𝑥 2 − 2𝑥 − 15 0 > (𝑥 − 5)(𝑥 + 3) turning points for sign (zeroes) are 𝑥 = 5 and 𝑥 = −3 [−4, 5[ 𝑓 𝑥 =2 𝑥+4−1 𝑔 𝑥 =𝑥 𝑓 𝑥 > 𝑔(𝑥) Ex 2 Vertex POI • Solve −2 9𝑥 < −12 • 𝑥≥0 9𝑥 > 6 9𝑥 > 36 𝑥>4 −2 9 4 = −12 −2 36 = −12 −2 6 = −12 −12 = −12 → 𝑇𝑅𝑈𝐸 Ex 3 ∴ 𝑥 ∈]4, ∞[ • Solve −2 𝑥 − 3 + 4 ≥ 6 • 𝑥≥3 −2 𝑥 − 3 ≥ 2 𝑥 − 3 ≤ −1 ∴ 𝑛𝑒𝑣𝑒𝑟 𝑡𝑜𝑢𝑐ℎ𝑒𝑠‼ Always or Never?? Check Visual Ex 4 −2 𝑥 − 3 + 4 ≤ 6 Solve − 1 4 − 𝑥 − 3 + 4 > −𝑥 • Restrictions 𝑥 ≤ 3 1 − − 𝑥 − 3 = −𝑥 − 4 4 − 𝑥 − 3 = 4𝑥 + 16 • Restrictions 4𝑥 + 16 ≥ 0 → 𝑥 ≥ −4 −𝑥 + 3 = 16𝑥 2 + 128𝑥 + 256 0 = 16𝑥 2 + 129𝑥 + 253 • Quad Form −129 ± 449 𝑥= 32 𝑥1 ≈ −3.369 𝑥2 ≈ −4.693 Ex 5 •Visions 1 •P39: #5acf •P42: #14ad Homework