Determine the number of integer values of x that satisfy the quadratic inequality 3x^2 - 1… — JAMB UTME

Determine the number of integer values of $x$ that satisfy the quadratic inequality $3x^2 - 14x - 5 < 0$ .

Answer: 5

The total number of integer values satisfying the inequality is 5.

Factoring

3x^2 - 14x - 5 < 0

gives

(3x + 1)(x - 5) < 0

. The critical roots are

x = -\frac{1}{3}

and

x = 5

. Since the parabola opens upward, the expression is negative between the roots, yielding the interval

-\frac{1}{3} < x < 5

. The integer values contained in this interval are

0, 1, 2, 3,

and

4

, which total 5 values.

Factor the quadratic expression

(3x + 1)(x - 5) < 0

Factoring identifies the critical values where the quadratic expression changes sign.

Determine the critical points

x = -\frac{1}{3}

and

x = 5

Setting each factor to zero gives the boundary roots of the equation.

Formulate the solution interval

-\frac{1}{3} < x < 5

Since the coefficient of

x^2

is positive, the quadratic curve is below the x-axis strictly between the two roots.

List and count the integer solutions

The integers are

0, 1, 2, 3, 4

, yielding 5 integer solutions.

Counting integers strictly greater than

-\frac{1}{3}

and strictly less than

5

Solving quadratic inequalities and finding integer solution counts

Estimated Time:1m 30s

Question