Q.$\dfrac{1}{2}\left(\dfrac{3x}{5} + 4\right) \ge \dfrac{1}{3}(x - 6)$

Linear Inequality Solutions – A First Look

Imagine you're standing on a number line. You know exactly where the number 5 is. But what if I asked you to stand on "all numbers greater than 5"? You can't stand on all of them at once — they stretch infinitely to the right. That's the core idea of an inequality: instead of one exact point, you get a whole region of possible values.

A linear inequality is just like a linear equation ($ax + b = 0$), but instead of an equals sign, you have one of these: $<$, $>$, $\le$, or $\ge$. The solution is not a single number — it's an interval (or a union of intervals) on the number line.

From Equation to Inequality

Start with a simple equation:

Solve it: $2x = 4 \implies x = 2$. One point.

Now change it to an inequality:

Solve it the same way — but the meaning changes. Subtract 3: $2x > 4$. Divide by 2: $x > 2$.

The solution is all numbers greater than 2. On a number line, you draw an open circle at 2 (because 2 itself is not included) and shade everything to the right.

The Four Types of Solutions

The solution set is usually written in interval notation:

• $x > 2$ → $(2, \infty)$
• $x \le -3$ → $(-\infty, -3]$

Parentheses $($ or $)$ mean the endpoint is not included. Brackets $[$ or $]$ mean it is included.

Solving a Linear Inequality: Step by Step

Solve $3x - 5 \le 7x + 3$.

1. Bring variable terms to one side:

   $3x - 5 - 7x \le 3$

   $-4x - 5 \le 3$

2. Isolate the variable term:

   $-4x \le 8$

3. Divide by the coefficient (here it's $-4$, so reverse the sign):

   $x \ge -2$

The solution is $x \ge -2$, or in interval notation: $[-2, \infty)$.

Why This Matters …