Equations and Inequalities
Here you solve linear equations, simultaneous equations, and quadratic equations, and you interpret inequalities on the number line. The unifying idea is that operations applied equally to both sides preserve equality, and that inequalities behave the same way except when multiplying or dividing by a negative number.
Core skills
- Solve linear equations by isolating the variable, clearing fractions by multiplying through by a common denominator.
- Solve simultaneous equations by substitution or elimination; elimination is usually faster when coefficients nearly match.
- Solve quadratics by factoring, completing the square, or the quadratic formula.
- Represent inequality solutions correctly, using open or closed endpoints.
Common traps
When you multiply or divide an inequality by a negative number, the inequality sign reverses: −2x>6 gives x<−3. Forgetting this is the single most common error. In simultaneous equations, watch for sign slips during subtraction of one equation from another. With quadratics, a solution set may have two roots; do not discard one. Also check that a fraction's denominator is never zero.
Worked example
Solve the system 2x+y=7 and x−y=2. Adding the equations eliminates y: 3x=9, so x=3. Substituting gives 3−y=2, so y=1. Verify in the first equation: 2(3)+1=7. Correct.
Rule of thumb: for inequalities, isolate the variable as usual, then reverse the sign only if you multiplied or divided by a negative.