Solve 2x + 3 > 7 and represent on number line

Question

Solve the inequality $2x + 3 > 7$ and represent the solution on a number line.

Solution — Step by Step

2x + 3 > 7

2x + 3 - 3 > 7 - 3

2x > 4

We subtracted 3 from both sides. This is valid because adding or subtracting the same number to both sides of an inequality preserves the direction of the inequality sign.

\frac{2x}{2} > \frac{4}{2}

x > 2

We divided by positive 2, so the inequality sign stays the same. Remember: dividing or multiplying by a negative number reverses the inequality sign. Since 2 is positive, no reversal here.

The solution is $x > 2$ , which means all real numbers greater than 2.

In set notation: $\{x \in \mathbb{R} : x > 2\}$

In interval notation: $(2, \infty)$

Draw a number line. Mark the point 2.

Since the inequality is strictly greater than (not $\geq$ ), we use an open circle at 2 (hollow dot), indicating 2 is not included in the solution.

Draw an arrow pointing to the right from 2, extending toward positive infinity.

The ray starting just after 2 going right represents all solutions.

←————○——————————→
   -1  0  1  2  3  4  5
              ○→→→→→→

The open circle at $x = 2$ and the arrow to the right is the complete representation.

Why This Works

Linear inequalities behave almost identically to linear equations — we isolate the variable using the same operations. The critical difference is what happens when we multiply or divide by a negative number: this flips the inequality direction because the number line reverses.

For example, $-2x > 4$ gives $x < -2$ (sign flips). This is because multiplying both sides by $-1$ turns “greater than” into “less than” — larger numbers become smaller numbers after negation.

Alternative Method

We can verify by substituting test values:

$x = 3$ : $2(3) + 3 = 9 > 7$ ✓ (in solution)
$x = 2$ : $2(2) + 3 = 7 > 7$ ✗ (not strictly greater — 7 is not greater than 7)
$x = 1$ : $2(1) + 3 = 5 > 7$ ✗ (not in solution)

This confirms $x > 2$ is correct.

For CBSE Class 11 boards: always write the solution in all three forms — inequality form ( $x > 2$ ), set notation, and interval notation. Also show the number line representation. Many students lose marks by omitting the number line or using a filled circle (closed dot) instead of an open circle for strict inequalities.

Common Mistake

The most common error is forgetting to flip the inequality when dividing by a negative number. If this question were $-2x + 3 > 7$ , we’d get $-2x > 4$ , so $x < -2$ (sign flips). Many students write $x > -2$ and lose all marks. Before dividing, always check the sign of the coefficient.

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

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