PRELIMINARY CONTEST 2024
Contest 28
St. John’s Grammar School – 38 pts (Winner)
Accra High School – 32 pts
St. Francis Girls’ SHS – 13 pts
Contest 29
Kumasi Academy – 56 pts (Winner)
Saviour SHS – 16 pts
Battor SHS – 14 pts
Contest 30
Ghana SHS, Koforidua – 49 pts (Winner)
Sefwi Wiawso SHS – 15 pts
Northern School of Business – 09 pts
ROUND 1
QUESTION 1
Solve the compound linear inequality $2x-5<3x+8<2x+5$.
ANSWER: $-13 < x < -3$
SOLUTION
Break into two inequalities:
Inequality 1:
$2x-5<3x+8$
$2x-3x<8+5$
$-x<13$
$x>-13$
Inequality 2:
$3x+8<2x+5$
$3x-2x<5-8$
$x<-3$
Combine the results:
$x>-13\text{ and }x<-3$
$-13<x<-3$
QUESTION 2
Solve the compound linear inequality $8-x<2x+2<11-x$.
ANSWER: $2 < x < 3$
SOLUTION
Break into two inequalities:
Inequality 1:
$8-x<2x+2$
$-x-2x<2-8$
$-3x<-6$
$\dfrac{-3x}{-3}>\dfrac{-6}{-3}$
$x>2$
Inequality 2:
$2x+2<11-x$
$2x+x<11-2$
$3x<9$
$\dfrac{3x}{3}<\dfrac{9}{3}$
$x<3$
Combine the results:
$x>2\text{ and }x<3$
$2<x<3$
QUESTION 3
Solve the compound linear inequality $x-12<3x-4<x+12$.
ANSWER: $-4 < x < 8$
SOLUTION
Break into two inequalities:
Inequality 1:
$x-12<3x-4$
$x-3x<-4+12$
$-2x<8$
$\dfrac{-2x}{-2}>\dfrac{8}{-2}$
$x>-4$
Inequality 2:
$3x-4<x+12$
$3x-x<12+4$
$2x<16$
$\dfrac{2x}{2}<\dfrac{16}{2}$
$x<8$
Combine the results:
$x>-4\text{ and }x<8$
$-4<x<8$
PRACTICE QUESTIONS
1. Solve the compound linear inequality $-2x+1\lt-3x+8\lt-2x+14$.
ANSWER: $-6\lt x\lt 7$
SOLUTION
Break into two inequalities.
Inequality 1:
$-2x+1\lt-3x+8$
$-2x+3x\lt8-1$
$x\lt 7$
Inequality 2:
$-3x+8\lt-2x+14$
$-3x+2x\lt14-8$
$-x\lt6$
Multiply by $-1$ and reverse the inequality sign:
$x\gt -6$
Combine the results:
$-6\lt x\lt 7$
2. Solve the compound linear inequality $-3x-11\lt-4x+3\lt-2x+8$.
ANSWER: $-\dfrac{5}{2}\lt x\lt 14$
SOLUTION
Break into two inequalities.
Inequality 1:
$-3x-11\lt-4x+3$
$-3x+4x\lt3+11$
$x\lt 14$
Inequality 2:
$-4x+3\lt-2x+8$
$-4x+2x\lt8-3$
$-2x\lt5$
Divide by $-2$ and reverse the inequality sign:
$x\gt -\dfrac{5}{2}$
Combine the results:
$-\dfrac{5}{2}\lt x\lt 14$
3. Solve the compound linear inequality $3x+3\ltx-2\lt2x+10$.
ANSWER: $-12\lt x\lt -\dfrac{5}{2}$
SOLUTION
Break into two inequalities.
Inequality 1:
$3x+3\ltx-2$
$3x-x\lt-2-3$
$2x\lt-5$
Divide by $2$:
$x\lt -\dfrac{5}{2}$
Inequality 2:
$x-2\lt2x+10$
$x-2x\lt10+2$
$-x\lt12$
Multiply by $-1$ and reverse the inequality sign:
$x\gt -12$
Combine the results:
$-12\lt x\lt -\dfrac{5}{2}$
4. Solve the compound linear inequality $-2x-10\ltx-1\lt-x-5$.
ANSWER: $-3\lt x\lt -2$
SOLUTION
Break into two inequalities.
Inequality 1:
$-2x-10\ltx-1$
$-2x-x\lt-1+10$
$-3x\lt9$
Divide by $-3$ and reverse the inequality sign:
$x\gt -3$
Inequality 2:
$x-1\lt-x-5$
$x+x\lt-5+1$
$2x\lt-4$
Divide by $2$:
$x\lt -2$
Combine the results:
$-3\lt x\lt -2$
5. Solve the compound linear inequality $3x-6\lt2x+12\lt3x-2$.
ANSWER: $14\lt x\lt 18$
SOLUTION
Break into two inequalities.
Inequality 1:
$3x-6\lt2x+12$
$3x-2x\lt12+6$
$x\lt 18$
Inequality 2:
$2x+12\lt3x-2$
$2x-3x\lt-2-12$
$-x\lt-14$
Multiply by $-1$ and reverse the inequality sign:
$x\gt 14$
Combine the results:
$14\lt x\lt 18$
6. Solve the compound linear inequality $-x+1\lt3x+1\lt2x+12$.
ANSWER: $0\lt x\lt 11$
SOLUTION
Break into two inequalities.
Inequality 1:
$-x+1\lt3x+1$
$-x-3x\lt1-1$
$-4x\lt0$
Divide by $-4$ and reverse the inequality sign:
$x\gt 0$
Inequality 2:
$3x+1\lt2x+12$
$3x-2x\lt12-1$
$x\lt 11$
Combine the results:
$0\lt x\lt 11$