Find the number of integer values of x that satisfy the inequality 3x^2 - 10x - 8 \leq 0. — JAMB UTME

Find the number of integer values of $x$ that satisfy the inequality $3x^2 - 10x - 8 \leq 0$ .

Cevap: 5

The number of integer values of

x

satisfying the inequality is 5.

Factoring

3x^2 - 10x - 8 \leq 0

gives

(3x + 2)(x - 4) \leq 0

. The region where the quadratic expression is non-positive lies between the roots

x = -\frac{2}{3}

and

x = 4

, yielding

-\frac{2}{3} \leq x \leq 4

. The integers falling within this closed interval are

0, 1, 2, 3,

and

4

, giving a total of 5 integer solutions.

Factor the quadratic expression

(3x + 2)(x - 4) \leq 0

Factoring allows us to find the critical boundary values of the inequality.

Find the critical values (roots)

x = -\frac{2}{3}

and

x = 4

Setting each linear factor to zero determines where the expression changes sign.

Determine the solution set interval

-\frac{2}{3} \leq x \leq 4

Since the coefficient of

x^2

is positive, the quadratic curve is convex (U-shaped), so the expression is less than or equal to zero between the roots.

List and count the integer solutions

Integers:

0, 1, 2, 3, 4

(Total = 5)

The smallest integer greater than or equal to

-\frac{2}{3}

0

, and the largest integer less than or equal to

4

4

Solving quadratic inequalities and identifying integer solutions within a continuous range.

Soru