Linear and Quadratic Inequalities Alıştırma Soruları — JAMB UTME — Sayfa 2

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

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

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.

Adım Adım Çözüm

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.

Anahtar Kavram

Solving linear inequalities with fractions and finding integer bounds

Soru 22Soru →

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

Cevabı ve açıklamayı göster

Cevap: 27

Cevap

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

.

Adım Adım Çözüm

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.

Anahtar Kavram

Linear and Quadratic Inequalities

Tahmini Süre:1m 30s

Soru 23Soru →

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

Cevabı ve açıklamayı göster

Cevap: $x \le 4$

Cevap

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

.

Adım Adım Çözüm

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.

Anahtar Kavram

Solving linear inequalities involving negative coefficient division

Soru 24Soru →

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

Cevabı ve açıklamayı göster

Cevap: $x \ge -1$

Cevap

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

.

Adım Adım Çözüm

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.

Anahtar Kavram

Linear Inequalities and Reversing Inequality Sign on Division by Negative Numbers

Soru 25Soru →

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

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

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.

Adım Adım Çözüm

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.

Anahtar Kavram

Solving Compound Linear Inequalities

Soru 26Soru →

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

Cevabı ve açıklamayı göster

Cevap: $x \ge -2$

Cevap

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

.

Adım Adım Çözüm

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.

Anahtar Kavram

Solving linear inequalities involving bracket expansion and division by negative numbers

Soru 27Soru →

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

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

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.

Adım Adım Çözüm

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 ]

.

Anahtar Kavram

Quadratic Inequalities and Integer Solution Counting

Linear and Quadratic Inequalities

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar