Solving linear equations and inequalitiesIB MYP Maths Extended: Revision notes

An equation says two expressions are equal. To solve it, do the same thing to both sides so it stays balanced, using inverse operations to undo what has been done to $x$. For $3x+5=20$: subtract 5 to get $3x=15$, then divide by 3 to get $x=5$. Always check by substituting back: $3(5)+5=20$.

When $x$ appears on both sides, collect the $x$ terms on one side and the numbers on the other. For $5x-7=2x+11$: subtract $2x$ from both sides to get $3x-7=11$, add 7 to get $3x=18$, so $x=6$. Check: left side $5(6)-7=23$, right side $2(6)+11=23$. Subtracting the smaller $x$ term keeps the coefficient positive.

Expand the brackets first, multiplying every term inside, then solve as before. $3(2x-1)=x+17$ becomes $6x-3=x+17$, so $5x=20$ and $x=4$. A negative outside the bracket changes both signs inside: $-2(x-3)=-2x+6$.

Remove fractions by multiplying every term by the lowest common denominator. $\frac{3x+1}{4}=\frac{x+5}{2}$: multiply both sides by 4 to get $3x+1=2(x+5)$, so $3x+1=2x+10$ and $x=9$. For $\frac{x}{2}+\frac{x-3}{4}=3$: multiply by 4 to get $2x+(x-3)=12$, so $3x=15$ and $x=5$. Keep the numerator in brackets when multiplying so the minus sign applies to the whole numerator.

An inequality compares expressions using $<$, $>$, $\le$ (less than or equal to) or $\ge$. Solve it like an equation, with one change: when you multiply or divide by a negative number, reverse the inequality sign. $4x+3<2x+13$ gives $2x<10$, so $x<5$. $7-3x\ge1$ gives $-3x\ge-6$, so $x\le2$. Check by testing a value: $x=0$ in $7-3x\ge1$ gives $7\ge1$, which is true, and $0\le2$.

On a number line, a solution set is shown by a line with an arrow. Use an open circle when the end value is not included ($<$ or $>$) and a filled circle when it is included ($\le$ or $\ge$). $x<5$ is an open circle at 5 with an arrow pointing left. $x\le2$ is a filled circle at 2 with an arrow pointing left. For whole-number solutions, list them: $x<8$ with $x\ge0$ gives $0,1,2,3,4,5,6,7$.