Find the sum of all integer values of x that satisfy both the linear inequality 2x - 1 \ge… — JAMB UTME

Find the sum of all integer values of $x$ that satisfy both the linear inequality $2x - 1 \ge 3$ and the quadratic inequality $x^2 - 5x - 14 \le 0$ .

Answer: 27

The sum of all integer values of

x

satisfying both inequalities is

27

Solving the linear inequality

2x - 1 \ge 3

yields

x \ge 2

. Solving the quadratic inequality

x^2 - 5x - 14 \le 0

by factoring gives

(x - 7)(x + 2) \le 0

, which defines the interval

-2 \le x \le 7

. Taking the intersection of

x \ge 2

and

-2 \le x \le 7

gives

2 \le x \le 7

. The integer values satisfying this range are

2, 3, 4, 5, 6,

and

7

, and their sum is

27

Solve the linear inequality

2x \ge 4 \implies x \ge 2

Adding 1 to both sides and dividing by 2 isolates the variable

x

Factor and solve the quadratic inequality

(x - 7)(x + 2) \le 0 \implies -2 \le x \le 7

The roots of the quadratic equation are

x = 7

and

x = -2

. The parabola opens upward, so the expression is non-positive between the roots.

Determine the intersection of both solution sets

2 \le x \le 7

The values of

x

must simultaneously satisfy

x \ge 2

and

-2 \le x \le 7

List all integer solutions within the valid interval

x \in \{2, 3, 4, 5, 6, 7\}

These are all the whole numbers contained in the closed interval

[2, 7]

Sum the integer solutions

2 + 3 + 4 + 5 + 6 + 7 = 27

Summing the identified integer values yields the final required numerical answer.

Linear and Quadratic Inequalities

Estimated Time:1m 30s

Question