1. Linear Inequality Solver – Theory
This section reviews how to solve linear inequalities in one variable \(x\),
including strict and non-strict inequalities, compound inequalities, number-line
representations, and how to translate common word-problem phrases into algebraic
inequalities.
1. What is a linear inequality?
A linear inequality in one variable has the general form
\[
a x + b \,\square\, c
\]
where \(a\), \(b\), \(c\) are real numbers (with \(a\neq 0\)) and
\(\square\) is one of the inequality symbols
\[
<, \; >, \; \le, \; \ge.
\]
- \(x < c\): “\(x\) is less than \(c\)” (strict).
- \(x \le c\): “\(x\) is less than or equal to \(c\)” (non-strict).
- \(x > c\): “\(x\) is greater than \(c\)” (strict).
- \(x \ge c\): “\(x\) is greater than or equal to \(c\)” (non-strict).
Unlike equations, which typically have a finite set of solutions, linear
inequalities usually describe an interval of real numbers.
2. Operations on inequalities
To solve a linear inequality, we use operations that transform the inequality
into an equivalent one (with the same solution set).
Key idea: the inequality sign flips only when you multiply or divide by a
negative number. The calculator highlights this step explicitly.
3. Solving a single linear inequality
Consider the typical inequality
\[
a x + b \,\square\, c.
\]
To solve it:
- Expand parentheses, e.g. distribute factors like \(5(2-x)\).
- Collect like terms: bring all \(x\)-terms to one side and constants to the other.
- Isolate \(x\) by dividing by the coefficient of \(x\), flipping the inequality sign if that coefficient is negative.
Example 1 – sample from the calculator
Solve
\[
-5(2 - x) + 3 \;\ge\; 8x - 7.
\]
In the last step we divided by \(-3\), so the inequality sign flipped
from \(\ge\) to \(\le\).
The solution set is \(\{x \in \mathbb{R} : x \le 0\}\), or in interval
notation \((-\infty, 0]\).
4. Compound linear inequalities
A compound inequality such as
\[
-3 < 2x + 1 \le 9
\]
combines two inequalities:
\[
-3 < 2x + 1
\quad\text{and}\quad
2x + 1 \le 9
\]
which must both be true at the same time.
Example 2 – compound inequality
The solution set is all real numbers \(x\) such that \(-2 < x \le 4\),
i.e. the open interval at \(-2\) and closed at \(4\).
The calculator does exactly this: it solves each inequality separately
and then takes the intersection of the two solution intervals.
5. Number line representation
Solutions of linear inequalities are often shown on a number line:
- Closed dot (●) at a point means the endpoint is included: \(\le\) or \(\ge\).
- Open dot (○) at a point means the endpoint is excluded: < or >.
- Shaded line indicates all real numbers that satisfy the inequality.
For example:
-
\(x \le 2\): closed dot at \(2\), shading to the left.
-
\(x > -1\): open dot at \(-1\), shading to the right.
-
\(-2 < x \le 4\): open dot at \(-2\), closed dot at \(4\), shading between them.
The calculator’s number line uses exactly this convention: closed dots for
\(\le,\ge\) and open dots for <,>.
6. Word problem mode: phrases → inequalities
Many real-life problems describe conditions using words instead of symbols.
The calculator’s “word problem mode” recognises several common phrases and
converts them into inequalities in \(x\).
Once the phrase is translated into an algebraic inequality, the solving
process is exactly the same as in the purely symbolic case.
7. Checklist for solving linear inequalities
- Check that the expression is linear (no \(x^2\), no \(x\) in denominators).
- Expand parentheses and simplify both sides.
- Move all \(x\)-terms to one side and constants to the other.
- Divide by the coefficient of \(x\); flip the sign if that coefficient is negative.
- For compound inequalities, solve both parts and intersect the intervals.
- Draw a number line with open/closed dots and shading to visualise the solution set.
