1 markMCQSection A

The real x for which $2(2x + 3) - 10 < 6(x - 2)$ is :

Q: The real x for which $2(2x + 3) - 10 < 6(x - 2)$ is :

The correct option is **C) $x > 4$**. **Working:** - Expand: $2(2x + 3) - 10 4$.

Algebra (Linear Inequalities)

Linear Inequalities

Options

(A)

x > 2

(B)

x > 3

(C)

x > 4

(D)

x > -4

Official Answer

The correct option is C) $x > 4$ .

Working:

Expand: $2(2x + 3) - 10 < 6(x - 2)$
$4x + 6 - 10 < 6x - 12$
$4x - 4 < 6x - 12$
Collect terms: $12 - 4 < 6x - 4x$
$8 < 2x$ , so $x > 4$ .

linear inequalitytranspositionx > 4one variablesolution setinequality directionpositive divisor

Marking Scheme

11 mark: correct simplification to $8 < 2x$ and final answer $x > 4$ (option C).

Hint

Expand both brackets, gather x-terms on one side; dividing by a positive number does not flip the inequality.

Quick Oral Answer

Expanding gives $8 < 2x$ , so $x > 4$ ; the sign stays the same because we divide by the positive number 2.

Analysis & Explanation

This tests solving a linear inequality in one variable by transposing terms.

Concept:

Expand both sides, bring variable terms to one side and constants to the other.
Dividing by a positive number (here +2) keeps the inequality direction unchanged.

Why C is correct:

$4x + 6 - 10 < 6x - 12$ simplifies to $8 < 2x$ , i.e. $x > 4$ .

Why the distractors are wrong:

A ( $x > 2$ ): arises from a sign slip while transposing the constants.
B ( $x > 3$ ): comes from mis-adding 6 - 10 as -3 instead of -4.
D ( $x > -4$ ): results from forgetting to move 6x across, or a sign error on the RHS constant.

Common Mistakes

1Flipping the inequality sign even though division is by a positive number (+2).
2Arithmetic slip in $6 - 10 = -4$ (often wrongly taken as -3 or +4).
3Failing to transpose $6x$ to the left, leaving a wrong constant-only inequality.

Interesting Facts

Linear inequalities are the algebraic backbone of Linear Programming — the feasible region in every LPP is just the overlap of several such inequalities.

Unlike equations, inequalities have infinitely many solutions, expressed as an interval — here $(4, \infty)$ .

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Frequently Asked Questions

When do we reverse the inequality sign?

Only when multiplying or dividing both sides by a negative number. Here we divide by +2, so the sign stays as it is and we get $x > 4$ .

How is the answer written as a set?

The solution is all real numbers greater than 4, written as the interval $(4, \infty)$ or $\{x \in R : x > 4\}$ .