Q.A man wants to cut three lengths from a single piece of board of length 91 cm. The second length is to be 3 cm longer than the shortest and the third length is to be twice as long as the shortest. What are the possible lengths of the shortest board if the third piece is to be at least 5 cm longer than the second? [Hint: If $x$ is the length of the shortest board, then $x$, $(x + 3)$ and $2x$ are the lengths of the second and third piece, respectively. Thus, $x + (x + 3) + 2x \le 91$ and $2x \ge (x + 3) + 5$].

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 …