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CH-08-001 · Intersecting quadratic and linear inequalities

Find the set of values of x satisfying both x2−4x−60<0x^{2} - 4x - 60<0 and 5x<145x<14. Show all your working.

Quadratic & Linear Inequalities (Intersection) Booklet p. 15 & p. 16
Method
1) Factorise quadratic to find critical values and sketch U-parabola (a < x < b)
2) Solve linear inequality (kill, kill)
3) Find overlapping region
Worked Example (CH-08-001)
1) Factorise to find critical values (Booklet p. 5 & p. 16):
×xx+6+6
xxx2x^2+6x+6x
−10-10−10x-10x−60-60
a×c=1×(−60)=−60a \times c = 1 \times (-60) = -60
+1+1−60-60
+2+2−30-30
+3+3−20-20
+4+4−15-15
+5+5−12-12
+6+6−10-10
c is negative so signs are different (+ and −)...
+6−10=−4+6 - 10 = -4 ✔
Brackets: (x−10)(x+6)\text{Brackets: } \mathbf{(x - 10)(x + 6)}
2) Critical values: x=−6x = -6 and x=10x = 10
3) Sketch graph: quadratic <0< 0 (below axis) ⇒−6<x<10\Rightarrow -6 < x < 10
4) Intersect with linear inequality: x<145x<\frac{14}{5}
-6 10
−6<x<145-6<x<\frac{14}{5}