Linear and Quadratic Inequalities Practice Questions — JAMB UTME — Page 2

Find the largest integer value of $x$ that satisfies the linear inequality $\frac{2x - 5}{3} \le \frac{x + 1}{4}$ .

Show answer & explanation

Answer: 4

Answer

The largest integer value of

x

satisfying the inequality is 4.

Multiplying the entire inequality by 12 yields 4(2x - 5) <= 3(x + 1). Expanding both sides produces 8x - 20 <= 3x + 3. Subtracting 3x and adding 20 gives 5x <= 23, which simplifies to x <= 4.6. The largest integer less than or equal to 4.6 is 4.

Step-by-Step Solution

1

Clear the denominators by multiplying both sides by 12.

4(2x - 5) \le 3(x + 1)

Multiplying by a positive number preserves the inequality direction while clearing fractions.

2

Expand both sides of the inequality using the distributive property.

8x - 20 \le 3x + 3

Multiply 4 through (2x - 5) and 3 through (x + 1).

3

Isolate the variable terms on one side and constant terms on the other.

5x \le 23 \implies x \le 4.6

Subtract 3x from both sides and add 20 to both sides, then divide by 5.

4

Determine the maximum integer value satisfying the inequality boundary.

4

Since x must be less than or equal to 4.6, the greatest whole integer satisfying this condition is 4.

Key Concept

Solving linear inequalities with fractions and finding integer bounds

Question 22Question →

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

Show answer & explanation

Answer: 27

Answer

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

.

Step-by-Step Solution

1

Solve the linear inequality

2x \ge 4 \implies x \ge 2

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

x

.

2

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.

3

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

.

4

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]

.

5

Sum the integer solutions

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

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

Key Concept

Linear and Quadratic Inequalities

Estimated Time:1m 30s

Question 23Question →

Find the set of real values of $x$ that satisfies the inequality $\frac{5 - 2x}{3} \ge x - 5$ .

x \ge 4

x \le 4

x \le -4

x \ge -4

Show answer & explanation

Answer: $x \le 4$

Answer

The set of real values satisfying the inequality is

x \le 4

.

Multiplying through by

3

yields

5 - 2x \ge 3x - 15

. Rearranging terms gives

-5x \ge -20

. Dividing both sides by

-5

requires flipping the inequality sign from

\ge

to

\le

, giving

x \le 4

.

Step-by-Step Solution

1

Multiply both sides of the inequality by 3 to clear the fraction.

5 - 2x \ge 3(x - 5)

Eliminating the denominator simplifies the algebraic expression.

2

Expand the right-hand side and collect terms containing

x

on one side and constants on the other.

5 - 2x \ge 3x - 15 \implies -2x - 3x \ge -15 - 5 \implies -5x \ge -20

Group like terms to isolate the variable

x

.

3

Divide both sides by

-5

and reverse the inequality sign.

x \le \frac{-20}{-5} \implies x \le 4

Dividing or multiplying an inequality by a negative number reverses the direction of the inequality sign.

Key Concept

Solving linear inequalities involving negative coefficient division

Question 24Question →

Which of the following is the set of real values of $x$ that satisfies the inequality $\frac{3 - x}{4} \le \frac{2x + 5}{3}$ ?

x \ge -1

x \le -1

x \ge 1

x \le 1

Show answer & explanation

Answer: $x \ge -1$

Answer

The set of real values of

x

that satisfies the inequality is

x \ge -1

.

Multiplying through by 12 gives

9 - 3x \le 8x + 20

. Grouping terms results in

-11x \le 11

. Dividing by

-11

requires reversing the inequality sign from

\le

to

\ge

, giving the solution

x \ge -1

.

Step-by-Step Solution

1

Clear the denominators by multiplying both sides of the inequality by the lowest common multiple, 12.

3(3 - x) \le 4(2x + 5)

Eliminating fractions simplifies the algebraic expression.

2

Expand both sides by distributing the multipliers.

9 - 3x \le 8x + 20

Prepares terms for grouping variables on one side and constants on the other.

3

Collect all terms containing

x

on the left side and constant terms on the right side.

-3x - 8x \le 20 - 9 \implies -11x \le 11

Isolates the linear variable term.

4

Divide both sides by

-11

and flip the inequality sign.

x \ge -1

Dividing or multiplying an inequality by a negative number reverses the direction of the inequality sign.

Key Concept

Linear Inequalities and Reversing Inequality Sign on Division by Negative Numbers

Question 25Question →

Determine the smallest integer value of $x$ that satisfies the compound inequality $3 < 2x - 5 \le 11$ .

Show answer & explanation

Answer: 5

Answer

The smallest integer value of

x

that satisfies the inequality is 5.

Adding 5 across the compound inequality

3 < 2x - 5 \le 11

gives

8 < 2x \le 16

. Dividing by 2 yields

4 < x \le 8

. The integer values satisfying this range are 5, 6, 7, and 8. Therefore, the smallest integer solution is 5.

Step-by-Step Solution

1

Add 5 to all parts of the compound inequality

8 < 2x <= 16

Isolates the variable term in the middle segment.

2

Divide all parts of the compound inequality by 2

4 < x <= 8

Solves for x without changing inequality signs since 2 is positive.

3

Identify integer solutions within the range (4, 8]

x in {5, 6, 7, 8}

Since the inequality at 4 is strict (<), 4 is excluded, but 8 is included (<=).

4

Find the minimum integer value in the solution set

5

5 is the smallest integer strictly greater than 4.

Key Concept

Solving Compound Linear Inequalities

Question 26Question →

Which of the following represents the solution set of real values of $x$ satisfying the inequality $4(1 - x) \le 3(x + 6)$ ?

x \ge -2

x \le -2

x \ge 2

x \le 2

Show answer & explanation

Answer: $x \ge -2$

Answer

The set of real values satisfying the inequality is

x \ge -2

.

Expanding the given inequality yields

4 - 4x \le 3x + 18

. Subtracting

3x

and

4

from both sides gives

-7x \le 14

. When dividing both sides by

-7

, the inequality sign must reverse direction, yielding

x \ge -2

.

Step-by-Step Solution

1

Expand both sides of the inequality

4 - 4x \le 3x + 18

Remove brackets to group like terms.

2

Rearrange terms by moving variable terms to the left side and constant terms to the right side

-4x - 3x \le 18 - 4 \implies -7x \le 14

Isolate the term containing the variable

x

.

3

Divide both sides by

-7

and reverse the inequality sign

x \ge -2

Dividing or multiplying an inequality by a negative number flips the direction of the inequality symbol.

Key Concept

Solving linear inequalities involving bracket expansion and division by negative numbers

Question 27Question →

How many integer values of $x$ satisfy the quadratic inequality $2x^2 - 7x - 4 \le 0$ ?

Show answer & explanation

Answer: 5

Answer

There are 5 integer values of x that satisfy the inequality.

Solving the quadratic inequality yields

-\frac{1}{2} \le x \le 4

. The integer solutions within this interval are

0, 1, 2, 3,

and

4

. Counting them gives a total of

5

valid integer values.

Step-by-Step Solution

1

Factor the quadratic expression.

(2x + 1)(x - 4) \le 0

Factoring allows determination of the critical boundary points.

2

Find the critical values (roots of the equation).

x = -\frac{1}{2}

and

x = 4

The roots divide the number line into test intervals.

3

Determine the solution set of the inequality.

-\frac{1}{2} \le x \le 4

Since the quadratic coefficient is positive, the quadratic expression is non-positive between its roots.

4

List and count the integers within the range.

The integers are

0, 1, 2, 3, 4

, making a total of

5

integers.

Counting only whole numbers in the closed interval

[ -0.5, 4 ]

.

Key Concept

Quadratic Inequalities and Integer Solution Counting

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