Solved Exercises on Systems of Inequalities: Learn how to solve linear and quadratic systems of inequalities using the sign chart, the intersection of solution sets and the correct way of writing intervals in set-builder notation.
Exercise 1 — level ★★☆☆☆
\[ \begin{cases} 2x - 4 > 0 \\ x - 3 < 0 \end{cases} \]
Result
\[ 2 < x < 3 \]
Solution
First inequality
\(2x-4>0 \implies x>2\)
Second inequality
\(x-3<0 \implies x<3\)
Sign chart
The Sys. row is green where both conditions are simultaneously satisfied.
Solution set
\[ S = (2,\,3) \]
Result
\[ \boxed{2 < x < 3} \]
Exercise 2 — level ★★☆☆☆
\[ \begin{cases} x + 2 > 0 \\ x - 5 < 0 \end{cases} \]
Result
\[ -2 < x < 5 \]
Solution
First inequality
\(x+2>0 \implies x>-2\)
Second inequality
\(x-5<0 \implies x<5\)
Sign chart
Solution set
\[ S = (-2,\,5) \]
Result
\[ \boxed{-2 < x < 5} \]
Exercise 3 — level ★★☆☆☆
\[ \begin{cases} 3x + 1 \geq 0 \\ 2x - 4 \leq 0 \end{cases} \]
Result
\[ -\dfrac{1}{3} \leq x \leq 2 \]
Solution
First inequality
\(3x+1\geq0 \implies x\geq-\tfrac{1}{3}\)
Second inequality
\(2x-4\leq0 \implies x\leq2\)
Sign chart
The filled circles indicate that the endpoints are included.
Solution set
\[ S = \left[-\tfrac{1}{3},\,2\right] \]
Result
\[ \boxed{-\dfrac{1}{3} \leq x \leq 2} \]
Exercise 4 — level ★★☆☆☆
\[ \begin{cases} x - 1 > 0 \\ x + 4 < 0 \end{cases} \]
Result
\[ \text{No solution} \quad (S = \emptyset) \]
Solution
First inequality
\(x-1>0 \implies x>1\)
Second inequality
\(x+4<0 \implies x<-4\)
Observation
The two conditions \(x>1\) and \(x<-4\) are incompatible: there is no real \(x\) that satisfies both at the same time.
Sign chart
The Sys. row is entirely grey: no region is a solution.
Result
\[ \boxed{S = \emptyset} \]
Exercise 5 — level ★★★☆☆
\[ \begin{cases} x^2 - 4 > 0 \\ x + 3 > 0 \end{cases} \]
Result
\[ -3 < x < -2 \quad \text{or} \quad x > 2 \]
Solution
First inequality: \(x^2-4>0\)
\[ (x-2)(x+2)>0 \implies x < -2 \;\text{ or }\; x>2 \]
Second inequality: \(x+3>0\)
\[ x > -3 \]
Sign chart
Solution set
\[ S = (-3,\,-2)\cup(2,\,+\infty) \]
Result
\[ \boxed{-3 < x < -2 \quad \text{or} \quad x > 2} \]
Exercise 6 — level ★★★☆☆
\[ \begin{cases} x^2 - 9 \leq 0 \\ x - 1 > 0 \end{cases} \]
Result
\[ 1 < x \leq 3 \]
Solution
First inequality: \(x^2-9\leq0\)
\[ (x-3)(x+3)\leq0 \implies -3\leq x\leq3 \]
Second inequality: \(x-1>0\)
\[ x > 1 \]
Sign chart
The filled circle at \(x=3\) shows that the right endpoint is included (from the first inequality, which is \(\leq\)).
Solution set
\[ S = (1,\,3] \]
Result
\[ \boxed{1 < x \leq 3} \]
Exercise 7 — level ★★★☆☆
\[ \begin{cases} x^2 - 3x + 2 > 0 \\ x - 2 < 0 \end{cases} \]
Result
\[ x < 1 \]
Solution
First inequality: \(x^2-3x+2>0\)
\[ (x-1)(x-2)>0 \implies x < 1 \;\text{ or }\; x>2 \]
Second inequality: \(x-2<0\)
\[ x < 2 \]
Sign chart
Intersection: \((x<1\text{ or }x>2)\cap(x<2) = x<1\).
Solution set
\[ S = (-\infty,\,1) \]
Result
\[ \boxed{x < 1} \]
Exercise 8 — level ★★★☆☆
\[ \begin{cases} x^2 - 5x + 6 \leq 0 \\ x^2 - 4 \geq 0 \end{cases} \]
Result
\[ 2 \leq x \leq 3 \]
Solution
First inequality: \(x^2-5x+6\leq0\)
\[ (x-2)(x-3)\leq0 \implies 2\leq x\leq3 \]
Second inequality: \(x^2-4\geq0\)
\[ (x-2)(x+2)\geq0 \implies x\leq-2 \;\text{ or }\; x\geq2 \]
Sign chart
Intersection: \([2,3]\cap(x\leq-2\text{ or }x\geq2)=[2,3]\).
Solution set
\[ S = [2,\,3] \]
Result
\[ \boxed{2 \leq x \leq 3} \]
Exercise 9 — level ★★★☆☆
\[ \begin{cases} 2x^2 - x - 1 > 0 \\ x^2 - 4x + 3 \leq 0 \end{cases} \]
Result
\[ 1 < x \leq 3 \]
Solution
First inequality: \(2x^2-x-1>0\)
\[ (2x+1)(x-1)>0 \implies x < -\tfrac{1}{2} \;\text{ or }\; x>1 \]
Second inequality: \(x^2-4x+3\leq0\)
\[ (x-1)(x-3)\leq0 \implies 1\leq x\leq3 \]
Sign chart
Intersection: \((x<-\tfrac{1}{2}\text{ or }x>1)\cap[1,3]=(1,3]\). The point \(x=1\) is excluded because the first inequality is strict.
Solution set
\[ S = (1,\,3] \]
Result
\[ \boxed{1 < x \leq 3} \]
Exercise 10 — level ★★★★☆
\[ \begin{cases} x^2 - x - 6 < 0 \\ x + 1 \geq 0 \end{cases} \]
Result
\[ -1 \leq x < 3 \]
Solution
First inequality: \(x^2-x-6<0\)
\[ (x-3)(x+2)<0 \implies -2 < x < 3 \]
Second inequality: \(x+1\geq0\)
\[ x \geq -1 \]
Sign chart
Intersection: \((-2,3)\cap[-1,+\infty)=[-1,3)\). The filled circle at \(x=-1\) is included (from the second inequality, \(\geq\)); \(x=3\) is excluded (from the first, strict).
Solution set
\[ S = [-1,\,3) \]
Result
\[ \boxed{-1 \leq x < 3} \]
Exercise 11 — level ★★★★☆
\[ \begin{cases} x^2 - 2x - 3 \geq 0 \\ x^2 + x - 6 < 0 \end{cases} \]
Result
\[ -3 < x \leq -1 \]
Solution
First inequality: \(x^2-2x-3\geq0\)
\[ (x-3)(x+1)\geq0 \implies x\leq-1 \;\text{ or }\; x\geq3 \]
Second inequality: \(x^2+x-6<0\)
\[ (x+3)(x-2)<0 \implies -3 < x < 2 \]
Sign chart
Intersection: \((x\leq-1\text{ or }x\geq3)\cap(-3,2)=(-3,-1]\). The filled circle at \(x=-1\) is included; \(x=-3\) is excluded (second inequality, strict).
Solution set
\[ S = (-3,\,-1] \]
Result
\[ \boxed{-3 < x \leq -1} \]
Exercise 12 — level ★★★★☆
\[ \begin{cases} \dfrac{x-1}{x+2} > 0 \\[6pt] x^2 - 9 < 0 \end{cases} \]
Result
\[ -3 < x < -2 \quad \text{or} \quad 1 < x < 3 \]
Solution
First inequality
\[ \frac{x-1}{x+2}>0 \implies x < -2 \;\text{ or }\; x>1 \quad (x\neq-2) \]
Second inequality: \(x^2-9<0\)
\[ (x-3)(x+3)<0 \implies -3 < x < 3 \]
Sign chart
Solution set
\[ S = (-3,\,-2)\cup(1,\,3) \]
Result
\[ \boxed{-3 < x < -2 \quad \text{or} \quad 1 < x < 3} \]
Exercise 13 — level ★★★★☆
\[ \begin{cases} (x-2)^2 > 0 \\ x^2 - 1 < 0 \end{cases} \]
Result
\[ -1 < x < 1 \]
Solution
First inequality: \((x-2)^2>0\)
A square is always \(\geq0\); it equals \(0\) only at \(x=2\). Therefore \((x-2)^2>0\) for every \(x\neq2\).
Second inequality: \(x^2-1<0\)
\[ (x-1)(x+1)<0 \implies -1 < x < 1 \]
Sign chart
The first condition is satisfied everywhere except at \(x=2\), which does not belong to \((-1,1)\) anyway. The intersection is therefore \((-1,1)\) itself.
Solution set
\[ S = (-1,\,1) \]
Result
\[ \boxed{-1 < x < 1} \]
Exercise 14 — level ★★★★☆
\[ \begin{cases} x^2 + x - 2 \geq 0 \\ x^2 - x - 6 \leq 0 \end{cases} \]
Result
\[ x = -2 \quad \text{or} \quad 1 \leq x \leq 3 \]
Solution
First inequality: \(x^2+x-2\geq0\)
\[ (x+2)(x-1)\geq0 \implies x\leq-2 \;\text{ or }\; x\geq1 \]
Second inequality: \(x^2-x-6\leq0\)
\[ (x-3)(x+2)\leq0 \implies -2\leq x\leq3 \]
Sign chart
Intersection: \((x\leq-2\text{ or }x\geq1)\cap[-2,3]=\{-2\}\cup[1,3]\).
The isolated point \(x=-2\) belongs to both sets: \(x=-2\leq-2\) ✓ and \(-2\leq-2\leq3\) ✓.
Solution set
\[ S = \{-2\}\cup[1,\,3] \]
Result
\[ \boxed{x=-2 \quad \text{or} \quad 1\leq x\leq3} \]
Exercise 15 — level ★★★★★
\[ \begin{cases} x^2 - 5x + 6 > 0 \\ x^2 - 4x + 3 < 0 \end{cases} \]
Result
\[ 1 < x < 2 \]
Solution
First inequality: \(x^2-5x+6>0\)
\[ (x-2)(x-3)>0 \implies x < 2 \;\text{ or }\; x>3 \]
Second inequality: \(x^2-4x+3<0\)
\[ (x-1)(x-3)<0 \implies 1 < x < 3 \]
Sign chart
Intersection: \((x<2\text{ or }x>3)\cap(1<x<3)=(1,2)\).
Solution set
\[ S = (1,\,2) \]
Result
\[ \boxed{1 < x < 2} \]
Exercise 16 — level ★★★★★
\[ \begin{cases} x(x-3) > 0 \\ (x-1)(x-4) \leq 0 \end{cases} \]
Result
\[ 3 < x \leq 4 \]
Solution
First inequality: \(x(x-3)>0\)
\[ x<0 \;\text{ or }\; x>3 \]
Second inequality: \((x-1)(x-4)\leq0\)
\[ 1\leq x\leq4 \]
Sign chart
Intersection: \((x<0\text{ or }x>3)\cap[1,4]=(3,4]\). The filled circle at \(x=4\) is included.
Solution set
\[ S = (3,\,4] \]
Result
\[ \boxed{3 < x \leq 4} \]
Exercise 17 — level ★★★★★
\[ \begin{cases} (x+2)(x-1) > 0 \\ (x-4)(x+1) < 0 \end{cases} \]
Result
\[ 1 < x < 4 \]
Solution
First inequality
\[ (x+2)(x-1)>0 \implies x < -2 \;\text{ or }\; x>1 \]
Second inequality
\[ (x-4)(x+1)<0 \implies -1 < x < 4 \]
Sign chart
Intersection: \((x<-2\text{ or }x>1)\cap(-1,4)=(1,4)\). Note that \((-\infty,-2)\cap(-1,4)=\emptyset\).
Solution set
\[ S = (1,\,4) \]
Result
\[ \boxed{1 < x < 4} \]
Exercise 18 — level ★★★★★
\[ \begin{cases} x^2 - 9 \geq 0 \\ x^2 - 4x - 5 < 0 \end{cases} \]
Result
\[ 3 \leq x < 5 \]
Solution
First inequality: \(x^2-9\geq0\)
\[ (x-3)(x+3)\geq0 \implies x\leq-3 \;\text{ or }\; x\geq3 \]
Second inequality: \(x^2-4x-5<0\)
\[ (x-5)(x+1)<0 \implies -1 < x < 5 \]
Sign chart
Intersection: \((x\leq-3\text{ or }x\geq3)\cap(-1,5)=[3,5)\). The filled circle at \(x=3\) is included; \(x=5\) is excluded.
Solution set
\[ S = [3,\,5) \]
Result
\[ \boxed{3 \leq x < 5} \]
Exercise 19 — level ★★★★★
\[ \begin{cases} |x - 2| < 3 \\ x^2 - 1 > 0 \end{cases} \]
Result
\[ 1 < x < 5 \]
Solution
First inequality: \(|x-2|<3\)
\[ -3 < x-2 < 3 \implies -1 < x < 5 \]
Equivalent to \((x+1)(x-5)<0\).
Second inequality: \(x^2-1>0\)
\[ (x-1)(x+1)>0 \implies x < -1 \;\text{ or }\; x>1 \]
Sign chart
Intersection: \((-1,5)\cap(x<-1\text{ or }x>1)=(1,5)\).
Solution set
\[ S = (1,\,5) \]
Result
\[ \boxed{1 < x < 5} \]
Exercise 20 — level ★★★★★
\[ \begin{cases} x^2 - x - 2 \leq 0 \\ x(x-3) > 0 \end{cases} \]
Result
\[ -1 \leq x < 0 \]
Solution
First inequality: \(x^2-x-2\leq0\)
\[ (x-2)(x+1)\leq0 \implies -1\leq x\leq2 \]
Second inequality: \(x(x-3)>0\)
\[ x<0 \;\text{ or }\; x>3 \]
Sign chart
Intersection: \([-1,2]\cap(x<0\text{ or }x>3)=[-1,0)\). The filled circle at \(x=-1\) is included (first inequality, \(\leq\)); \(x=0\) is excluded (second, strict).
Solution set
\[ S = [-1,\,0) \]
Result
\[ \boxed{-1 \leq x < 0} \]