Linear Inequalities Class 11: notes and important questions

Linear Inequalities: revision notes

1. Inequalities

Statements with $<,>,\le ,\ge$ are inequalities. $ax+b<0$ (and the other three forms) with $a\ne 0$ is a linear inequality in one variable. The solution set is the set of all values of the variable that make it true; it depends on the set the variable comes from ($\mathbb{N}$, $\mathbb{Z}$ or $\mathbb{R}$).

2. Rules for solving

• The same number may be added to or subtracted from both sides.
• Both sides may be multiplied or divided by the same positive number.
• Multiplying or dividing both sides by a negative number reverses the sign: $-2x\ge 6\Rightarrow x\le -3$.
• Clear fractions by multiplying by the (positive) LCM of the denominators.

3. The number line

A solid dot marks an end-point that is included ($\le ,\ge$), an open circle one that is not ($<,>$). In interval notation: $x<4$ is $(-\mathrm{\infty },4)$; $-1\le x<3$ is $[-1,3)$.

4. Systems and double inequalities

Solve each inequality separately and take the values common to all (the intersection). A double inequality $a<px+q\le b$ can be solved all at once by doing the same operation to the three parts.

5. Word problems

Translate "at least" as $\ge$, "at most" / "not more than" as $\le$, "more than" as $>$. Check whether the answer must be a whole number (people, items) and state the final set of values.

Worked example 1

Solve $5-2x\le 11$, $x\in \mathbb{R}$.

$-2x\le 6\Rightarrow x\ge -3$: $[-3,\mathrm{\infty })$.

Worked example 2

Solve $-5\le {\displaystyle \frac{2x-1}{3}}<3$.

$-15\le 2x-1<9\Rightarrow -14\le 2x<10\Rightarrow -7\le x<5$.

Worked example 3

Solve $2x+3>7$ for $x\in \mathbb{N}$.

$x>2$: $\{3,4,5,\dots \}$.

Common errors

• Not reversing the sign when dividing by a negative number.
• Wrong end-points in the answer: $<$ gives an open circle and a round bracket.
• For a system, giving the union instead of the intersection.
• Giving a decimal number of people: round in the direction the inequality allows.

Exam tips

• Write the final answer as an interval (or a set for $\mathbb{N},\mathbb{Z}$) and draw it on a number line when asked.
• In word problems, write the inequality in words first, then in symbols.