Find the sum of all integer values of x that satisfy the system of inequalities 3x + 1 > 7… — JAMB UTME

Find the sum of all integer values of $x$ that satisfy the system of inequalities $3x + 1 > 7$ and $x^2 - 4x - 21 \le 0$ .

Answer: 25

Solving

3x + 1 > 7

gives

x > 2

. Solving

x^2 - 4x - 21 \le 0

gives

-3 \le x \le 7

. The combined condition is

2 < x \le 7

. The integer values satisfying this condition are 3, 4, 5, 6, and 7. The sum of these integers is

3 + 4 + 5 + 6 + 7 = 25

Solve the linear inequality

3x + 1 > 7

x > 2

Subtracting 1 from both sides gives

3x > 6

, and dividing by 3 yields

x > 2

Solve the quadratic inequality

x^2 - 4x - 21 \le 0

-3 \le x \le 7

Factoring the quadratic expression gives

(x - 7)(x + 3) \le 0

. The roots are

-3

and

7

, and the quadratic curve is non-positive between these roots.

Determine the intersection of the solution sets

x > 2

and

-3 \le x \le 7

2 < x \le 7

The lower bound is determined by the strict linear constraint

x > 2

, while the upper bound is determined by the inclusive quadratic constraint

x \le 7

List the integer values of

x

within

2 < x \le 7

and compute their sum.

Integers are

3, 4, 5, 6, 7

; their sum is

25

The strict inequality excludes

2

, while the inclusive inequality includes

7

System of linear and quadratic inequalities

Question