{"id":251,"date":"2026-09-08T13:38:01","date_gmt":"2026-09-08T13:38:01","guid":{"rendered":"https:\/\/repstaq.com\/study-guide-docs\/algebraic-expressions-equations-and-inequalities-g11-maths-t4p1-2\/"},"modified":"2026-09-08T13:38:01","modified_gmt":"2026-09-08T13:38:01","slug":"algebraic-expressions-equations-and-inequalities-g11-maths-t4p1-2","status":"publish","type":"post","link":"https:\/\/repstaq.com\/study-guide-docs\/algebraic-expressions-equations-and-inequalities-g11-maths-t4p1-2\/","title":{"rendered":"Algebraic expressions, equations and inequalities-G11-maths-T4P1"},"content":{"rendered":"\n

Algebraic Expressions, Equations and Inequalities<\/h2>\n\n\n\n

Algebraic expressions contain variables, constants and operations. An equation states that two expressions are equal, while an inequality compares expressions. In examinations, show all algebraic steps, state restrictions where necessary and check solutions that may have been introduced by squaring.<\/p>\n\n\n\n

Exponents and surds<\/h2>\n\n\n\n

An exponent shows how many times a base is used as a factor. Negative exponents represent reciprocals, while fractional exponents represent roots.<\/p>\n\n\n\n

a<\/mi>\u2212<\/mo>n<\/mi><\/mrow><\/msup>=<\/mo>1<\/mn>a<\/mi>n<\/mi><\/msup><\/mfrac><\/mrow><\/math><\/div>\n\n\n\n
a<\/mi>1<\/mn>n<\/mi><\/mfrac><\/msup>=<\/mo>a<\/mi>n<\/mi><\/mroot><\/mrow><\/math><\/div>\n\n\n\n
a<\/mi>m<\/mi>n<\/mi><\/mfrac><\/msup>=<\/mo>a<\/mi>m<\/mi><\/msup>n<\/mi><\/mroot><\/mrow><\/math><\/div>\n\n\n\n

A surd is an irrational root that cannot be simplified to a rational number. Examples include:<\/p>\n\n\n\n

2<\/mn><\/msqrt>,<\/mo>5<\/mn><\/msqrt>,<\/mo>3<\/mn>7<\/mn><\/msqrt><\/mrow><\/math><\/div>\n\n\n\n

A square root is defined only when its radicand is non-negative in the real number system.<\/p>\n\n\n\n

Laws of exponents<\/h2>\n\n\n\n

For non-zero bases, apply the following laws:<\/p>\n\n\n\n

a<\/mi>m<\/mi><\/msup>\u00d7<\/mo>a<\/mi>n<\/mi><\/msup>=<\/mo>a<\/mi>m<\/mi>+<\/mo>n<\/mi><\/mrow><\/msup><\/mrow><\/math><\/div>\n\n\n\n
a<\/mi>m<\/mi><\/msup>a<\/mi>n<\/mi><\/msup><\/mfrac>=<\/mo>a<\/mi>m<\/mi>\u2212<\/mo>n<\/mi><\/mrow><\/msup><\/mrow><\/math><\/div>\n\n\n\n
(<\/mo>a<\/mi>m<\/mi><\/msup>)<\/mo><\/mrow>n<\/mi><\/msup>=<\/mo>a<\/mi>m<\/mi>n<\/mi><\/mrow><\/msup><\/mrow><\/math><\/div>\n\n\n\n
(<\/mo>a<\/mi>b<\/mi>)<\/mo><\/mrow>n<\/mi><\/msup>=<\/mo>a<\/mi>n<\/mi><\/msup>b<\/mi>n<\/mi><\/msup><\/mrow><\/math><\/div>\n\n\n\n
(<\/mo>a<\/mi>b<\/mi><\/mfrac>)<\/mo><\/mrow>n<\/mi><\/msup>=<\/mo>a<\/mi>n<\/mi><\/msup>b<\/mi>n<\/mi><\/msup><\/mfrac><\/mrow><\/math><\/div>\n\n\n\n
a<\/mi>0<\/mn><\/msup>=<\/mo>1<\/mn><\/mrow><\/math><\/div>\n\n\n\n

Example<\/strong><\/p>\n\n\n\n

Simplify the expression.<\/p>\n\n\n\n

6<\/mn>x<\/mi>5<\/mn><\/msup>y<\/mi>\u2212<\/mo>2<\/mn><\/mrow><\/msup><\/mrow>3<\/mn>x<\/mi>2<\/mn><\/msup>y<\/mi>\u2212<\/mo>4<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><\/math><\/div>\n\n\n\n
=<\/mo>2<\/mn>x<\/mi>5<\/mn>\u2212<\/mo>2<\/mn><\/mrow><\/msup>y<\/mi>\u2212<\/mo>2<\/mn>\u2212<\/mo>(<\/mo>\u2212<\/mo>4<\/mn>)<\/mo><\/mrow><\/msup><\/mrow><\/math><\/div>\n\n\n\n
=<\/mo>2<\/mn>x<\/mi>3<\/mn><\/msup>y<\/mi>2<\/mn><\/msup><\/mrow><\/math><\/div>\n\n\n\n

Common Mistake<\/strong><\/p>\n\n\n\n

Do not apply exponent laws to addition. In general:<\/p>\n\n\n\n

a<\/mi>m<\/mi><\/msup>+<\/mo>a<\/mi>n<\/mi><\/msup>\u2260<\/mo>a<\/mi>m<\/mi>+<\/mo>n<\/mi><\/mrow><\/msup><\/mrow><\/math><\/div>\n\n\n\n

Simplifying expressions involving surds<\/h2>\n\n\n\n

To simplify a surd, identify a perfect-square factor.<\/p>\n\n\n\n

Example<\/strong><\/p>\n\n\n\n

Simplify the surd.<\/p>\n\n\n\n

72<\/mn><\/msqrt><\/math><\/div>\n\n\n\n
=<\/mo>36<\/mn>\u00d7<\/mo>2<\/mn><\/mrow><\/msqrt><\/mrow><\/math><\/div>\n\n\n\n
=<\/mo>6<\/mn>2<\/mn><\/msqrt><\/mrow><\/math><\/div>\n\n\n\n

Only like surds may be added or subtracted.<\/p>\n\n\n\n

3<\/mn>5<\/mn><\/msqrt>+<\/mo>2<\/mn>5<\/mn><\/msqrt>=<\/mo>5<\/mn>5<\/mn><\/msqrt><\/mrow><\/math><\/div>\n\n\n\n

To remove a surd from a denominator, rationalise the denominator.<\/p>\n\n\n\n

Example<\/strong><\/p>\n\n\n\n

4<\/mn>3<\/mn><\/msqrt><\/mfrac><\/math><\/div>\n\n\n\n
=<\/mo>4<\/mn>3<\/mn><\/msqrt><\/mfrac>\u00d7<\/mo>3<\/mn><\/msqrt>3<\/mn><\/msqrt><\/mfrac><\/mrow><\/math><\/div>\n\n\n\n
=<\/mo>4<\/mn>3<\/mn><\/msqrt><\/mrow>3<\/mn><\/mfrac><\/mrow><\/math><\/div>\n\n\n\n

For a denominator containing two terms, multiply by the conjugate.<\/p>\n\n\n\n

(<\/mo>a<\/mi>+<\/mo>b<\/mi>)<\/mo>(<\/mo>a<\/mi>\u2212<\/mo>b<\/mi>)<\/mo>=<\/mo>a<\/mi>2<\/mn><\/msup>\u2212<\/mo>b<\/mi>2<\/mn><\/msup><\/mrow><\/math><\/div>\n\n\n\n
1<\/mn>2<\/mn>+<\/mo>3<\/mn><\/msqrt><\/mrow><\/mfrac>\u00d7<\/mo>2<\/mn>\u2212<\/mo>3<\/mn><\/msqrt><\/mrow>2<\/mn>\u2212<\/mo>3<\/mn><\/msqrt><\/mrow><\/mfrac><\/mrow><\/math><\/div>\n\n\n\n
=<\/mo>2<\/mn>\u2212<\/mo>3<\/mn><\/msqrt><\/mrow>4<\/mn>\u2212<\/mo>3<\/mn><\/mrow><\/mfrac><\/mrow><\/math><\/div>\n\n\n\n
=<\/mo>2<\/mn>\u2212<\/mo>3<\/mn><\/msqrt><\/mrow><\/math><\/div>\n\n\n\n

Equations involving surds<\/h2>\n\n\n\n

Isolate the surd before squaring both sides. Squaring can introduce extraneous solutions, so substitute answers into the original equation.<\/p>\n\n\n\n

Example<\/strong><\/p>\n\n\n\n

Solve the equation.<\/p>\n\n\n\n

2<\/mn>x<\/mi>+<\/mo>3<\/mn><\/mrow><\/msqrt>=<\/mo>x<\/mi><\/mrow><\/math><\/div>\n\n\n\n

Since a square root is non-negative:<\/p>\n\n\n\n

x<\/mi>\u2265<\/mo>0<\/mn><\/mrow><\/math><\/div>\n\n\n\n

Square both sides.<\/p>\n\n\n\n

2<\/mn>x<\/mi>+<\/mo>3<\/mn>=<\/mo>x<\/mi>2<\/mn><\/msup><\/mrow><\/math><\/div>\n\n\n\n
x<\/mi>2<\/mn><\/msup>\u2212<\/mo>2<\/mn>x<\/mi>\u2212<\/mo>3<\/mn>=<\/mo>0<\/mn><\/mrow><\/math><\/div>\n\n\n\n
(<\/mo>x<\/mi>\u2212<\/mo>3<\/mn>)<\/mo>(<\/mo>x<\/mi>+<\/mo>1<\/mn>)<\/mo>=<\/mo>0<\/mn><\/mrow><\/math><\/div>\n\n\n\n
x<\/mi>=<\/mo>3<\/mn>or<\/mtext>x<\/mi>=<\/mo>\u2212<\/mo>1<\/mn><\/mrow><\/math><\/div>\n\n\n\n

Check the possible solutions. The value negative one does not satisfy the original equation. Therefore:<\/p>\n\n\n\n

x<\/mi>=<\/mo>3<\/mn><\/mrow><\/math><\/div>\n\n\n\n

Quadratic equations<\/h2>\n\n\n\n

A quadratic equation has the standard form:<\/p>\n\n\n\n

a<\/mi>x<\/mi>2<\/mn><\/msup>+<\/mo>b<\/mi>x<\/mi>+<\/mo>c<\/mi>=<\/mo>0<\/mn>,<\/mo>a<\/mi>\u2260<\/mo>0<\/mn><\/mrow><\/math><\/div>\n\n\n\n

Quadratic equations can be solved by factorisation, completing the square or using the quadratic formula. Always write the equation in standard form before choosing a method.<\/p>\n\n\n\n

The solutions are also called roots or zeros. Graphically, real roots are the x-coordinates of the x-intercepts of the corresponding parabola.<\/p>\n\n\n\n

Factorisation<\/h2>\n\n\n\n

Factorisation rewrites a quadratic expression as a product of two factors. Use the zero-product property:<\/p>\n\n\n\n

A<\/mi>B<\/mi>=<\/mo>0<\/mn>\u21d2<\/mo>A<\/mi>=<\/mo>0<\/mn>or<\/mtext>B<\/mi>=<\/mo>0<\/mn><\/mrow><\/math><\/div>\n\n\n\n

Example<\/strong><\/p>\n\n\n\n

Solve by factorisation.<\/p>\n\n\n\n

2<\/mn>x<\/mi>2<\/mn><\/msup>\u2212<\/mo>x<\/mi>\u2212<\/mo>6<\/mn>=<\/mo>0<\/mn><\/mrow><\/math><\/div>\n\n\n\n

Split the middle term.<\/p>\n\n\n\n

2<\/mn>x<\/mi>2<\/mn><\/msup>+<\/mo>3<\/mn>x<\/mi>\u2212<\/mo>4<\/mn>x<\/mi>\u2212<\/mo>6<\/mn>=<\/mo>0<\/mn><\/mrow><\/math><\/div>\n\n\n\n
x<\/mi>(<\/mo>2<\/mn>x<\/mi>+<\/mo>3<\/mn>)<\/mo>\u2212<\/mo>2<\/mn>(<\/mo>2<\/mn>x<\/mi>+<\/mo>3<\/mn>)<\/mo>=<\/mo>0<\/mn><\/mrow><\/math><\/div>\n\n\n\n
(<\/mo>2<\/mn>x<\/mi>+<\/mo>3<\/mn>)<\/mo>(<\/mo>x<\/mi>\u2212<\/mo>2<\/mn>)<\/mo>=<\/mo>0<\/mn><\/mrow><\/math><\/div>\n\n\n\n
2<\/mn>x<\/mi>+<\/mo>3<\/mn>=<\/mo>0<\/mn>or<\/mtext>x<\/mi>\u2212<\/mo>2<\/mn>=<\/mo>0<\/mn><\/mrow><\/math><\/div>\n\n\n\n
x<\/mi>=<\/mo>\u2212<\/mo>3<\/mn>2<\/mn><\/mfrac>or<\/mtext>x<\/mi>=<\/mo>2<\/mn><\/mrow><\/math><\/div>\n\n\n\n

Remember to look for a highest common factor before using other factorisation methods.<\/p>\n\n\n\n

Completing the square<\/h2>\n\n\n\n

Completing the square changes a quadratic expression into the form:<\/p>\n\n\n\n

a<\/mi>(<\/mo>x<\/mi>\u2212<\/mo>p<\/mi>)<\/mo><\/mrow>2<\/mn><\/msup>+<\/mo>q<\/mi><\/mrow><\/math><\/div>\n\n\n\n

Example<\/strong><\/p>\n\n\n\n

Solve by completing the square.<\/p>\n\n\n\n

x<\/mi>2<\/mn><\/msup>\u2212<\/mo>6<\/mn>x<\/mi>+<\/mo>5<\/mn>=<\/mo>0<\/mn><\/mrow><\/math><\/div>\n\n\n\n

Move the constant term.<\/p>\n\n\n\n

x<\/mi>2<\/mn><\/msup>\u2212<\/mo>6<\/mn>x<\/mi>=<\/mo>\u2212<\/mo>5<\/mn><\/mrow><\/math><\/div>\n\n\n\n

Add the square of half the coefficient of the linear term to both sides.<\/p>\n\n\n\n

x<\/mi>2<\/mn><\/msup>\u2212<\/mo>6<\/mn>x<\/mi>+<\/mo>9<\/mn>=<\/mo>\u2212<\/mo>5<\/mn>+<\/mo>9<\/mn><\/mrow><\/math><\/div>\n\n\n\n
(<\/mo>x<\/mi>\u2212<\/mo>3<\/mn>)<\/mo><\/mrow>2<\/mn><\/msup>=<\/mo>4<\/mn><\/mrow><\/math><\/div>\n\n\n\n
x<\/mi>\u2212<\/mo>3<\/mn>=<\/mo>\u00b1<\/mo>2<\/mn><\/mrow><\/math><\/div>\n\n\n\n
x<\/mi>=<\/mo>3<\/mn>\u00b1<\/mo>2<\/mn><\/mrow><\/math><\/div>\n\n\n\n
x<\/mi>=<\/mo>1<\/mn>or<\/mtext>x<\/mi>=<\/mo>5<\/mn><\/mrow><\/math><\/div>\n\n\n\n

Exam Tip<\/strong><\/p>\n\n\n\n

When the coefficient of the squared term is not one, first divide every term by that coefficient or factor it out.<\/p>\n\n\n\n\n

\n\n\"\"\n<\/a>\n<\/figure>\n\n\n\n

Quadratic formula<\/h2>\n\n\n\n

The quadratic formula solves any quadratic equation in standard form.<\/p>\n\n\n\n

x<\/mi>=<\/mo>\u2212<\/mo>b<\/mi>\u00b1<\/mo>b<\/mi>2<\/mn><\/msup>\u2212<\/mo>4<\/mn>a<\/mi>c<\/mi><\/mrow><\/msqrt><\/mrow>2<\/mn>a<\/mi><\/mrow><\/mfrac><\/mrow><\/math><\/div>\n\n\n\n

Example<\/strong><\/p>\n\n\n\n

Solve using the quadratic formula.<\/p>\n\n\n\n

2<\/mn>x<\/mi>2<\/mn><\/msup>+<\/mo>3<\/mn>x<\/mi>\u2212<\/mo>1<\/mn>=<\/mo>0<\/mn><\/mrow><\/math><\/div>\n\n\n\n

Identify the coefficients.<\/p>\n\n\n\n

a<\/mi>=<\/mo>2<\/mn>,<\/mo>b<\/mi>=<\/mo>3<\/mn>,<\/mo>c<\/mi>=<\/mo>\u2212<\/mo>1<\/mn><\/mrow><\/math><\/div>\n\n\n\n

Substitute carefully.<\/p>\n\n\n\n

x<\/mi>=<\/mo>\u2212<\/mo>3<\/mn>\u00b1<\/mo>3<\/mn>2<\/mn><\/msup>\u2212<\/mo>4<\/mn>(<\/mo>2<\/mn>)<\/mo>(<\/mo>\u2212<\/mo>1<\/mn>)<\/mo><\/mrow><\/msqrt><\/mrow>2<\/mn>(<\/mo>2<\/mn>)<\/mo><\/mrow><\/mfrac><\/mrow><\/math><\/div>\n\n\n\n
x<\/mi>=<\/mo>\u2212<\/mo>3<\/mn>\u00b1<\/mo>9<\/mn>+<\/mo>8<\/mn><\/mrow><\/msqrt><\/mrow>4<\/mn><\/mfrac><\/mrow><\/math><\/div>\n\n\n\n
x<\/mi>=<\/mo>\u2212<\/mo>3<\/mn>\u00b1<\/mo>17<\/mn><\/msqrt><\/mrow>4<\/mn><\/mfrac><\/mrow><\/math><\/div>\n\n\n\n

Leave exact answers in surd form unless a decimal approximation is requested.<\/p>\n\n\n\n

Quadratic inequalities<\/h2>\n\n\n\n

A quadratic inequality may contain signs such as less than, greater than, less than or equal to, or greater than or equal to. First find the critical values by solving the related quadratic equation. Then use a sign table or a sketch of the parabola.<\/p>\n\n\n\n

Example<\/strong><\/p>\n\n\n\n

Solve the inequality.<\/p>\n\n\n\n

x<\/mi>2<\/mn><\/msup>\u2212<\/mo>x<\/mi>\u2212<\/mo>6<\/mn>\u2264<\/mo>0<\/mn><\/mrow><\/math><\/div>\n\n\n\n

Factorise.<\/p>\n\n\n\n

(<\/mo>x<\/mi>\u2212<\/mo>3<\/mn>)<\/mo>(<\/mo>x<\/mi>+<\/mo>2<\/mn>)<\/mo>\u2264<\/mo>0<\/mn><\/mrow><\/math><\/div>\n\n\n\n

Find the critical values.<\/p>\n\n\n\n

x<\/mi>=<\/mo>\u2212<\/mo>2<\/mn>or<\/mtext>x<\/mi>=<\/mo>3<\/mn><\/mrow><\/math><\/div>\n\n\n\n

The parabola opens upwards, so the expression is non-positive between the roots. The equality sign means the endpoints are included.<\/p>\n\n\n\n

\u2212<\/mo>2<\/mn>\u2264<\/mo>x<\/mi>\u2264<\/mo>3<\/mn><\/mrow><\/math><\/div>\n\n\n\n

If the inequality is strict, the endpoints are excluded.<\/p>\n\n\n\n

Simultaneous equations<\/h2>\n\n\n\n

Simultaneous equations are equations that must be satisfied by the same values of the variables. Use substitution or elimination.<\/p>\n\n\n\n

Example<\/strong><\/p>\n\n\n\n

Solve the system.<\/p>\n\n\n\n

y<\/mi>=<\/mo>x<\/mi>+<\/mo>1<\/mn><\/mrow><\/math><\/div>\n\n\n\n
x<\/mi>2<\/mn><\/msup>+<\/mo>y<\/mi>=<\/mo>7<\/mn><\/mrow><\/math><\/div>\n\n\n\n

Substitute the first equation into the second.<\/p>\n\n\n\n

x<\/mi>2<\/mn><\/msup>+<\/mo>x<\/mi>+<\/mo>1<\/mn>=<\/mo>7<\/mn><\/mrow><\/math><\/div>\n\n\n\n
x<\/mi>2<\/mn><\/msup>+<\/mo>x<\/mi>\u2212<\/mo>6<\/mn>=<\/mo>0<\/mn><\/mrow><\/math><\/div>\n\n\n\n
(<\/mo>x<\/mi>+<\/mo>3<\/mn>)<\/mo>(<\/mo>x<\/mi>\u2212<\/mo>2<\/mn>)<\/mo>=<\/mo>0<\/mn><\/mrow><\/math><\/div>\n\n\n\n
x<\/mi>=<\/mo>\u2212<\/mo>3<\/mn>or<\/mtext>x<\/mi>=<\/mo>2<\/mn><\/mrow><\/math><\/div>\n\n\n\n

Calculate the corresponding values of the second variable.<\/p>\n\n\n\n

y<\/mi>=<\/mo>\u2212<\/mo>3<\/mn>+<\/mo>1<\/mn>=<\/mo>\u2212<\/mo>2<\/mn><\/mrow><\/math><\/div>\n\n\n\n
y<\/mi>=<\/mo>2<\/mn>+<\/mo>1<\/mn>=<\/mo>3<\/mn><\/mrow><\/math><\/div>\n\n\n\n

The solutions are:<\/p>\n\n\n\n

(<\/mo>x<\/mi>,<\/mo>y<\/mi>)<\/mo>=<\/mo>(<\/mo>\u2212<\/mo>3<\/mn>,<\/mo>\u2212<\/mo>2<\/mn>)<\/mo>or<\/mtext>(<\/mo>x<\/mi>,<\/mo>y<\/mi>)<\/mo>=<\/mo>(<\/mo>2<\/mn>,<\/mo>3<\/mn>)<\/mo><\/mrow><\/math><\/div>\n\n\n\n

On the Cartesian plane, these solutions represent the points of intersection of the graphs.<\/p>\n\n\n\n

Linear inequalities<\/h2>\n\n\n\n

Solve a linear inequality in the same way as a linear equation. However, reverse the inequality sign when multiplying or dividing by a negative number.<\/p>\n\n\n\n

Example<\/strong><\/p>\n\n\n\n

3<\/mn>x<\/mi>\u2212<\/mo>5<\/mn><<\/mo>10<\/mn><\/mrow><\/math><\/div>\n\n\n\n
3<\/mn>x<\/mi><<\/mo>15<\/mn><\/mrow><\/math><\/div>\n\n\n\n
x<\/mi><<\/mo>5<\/mn><\/mrow><\/math><\/div>\n\n\n\n

Example involving a negative coefficient:<\/strong><\/p>\n\n\n\n

\u2212<\/mo>2<\/mn>x<\/mi>+<\/mo>3<\/mn>\u2265<\/mo>11<\/mn><\/mrow><\/math><\/div>\n\n\n\n
\u2212<\/mo>2<\/mn>x<\/mi>\u2265<\/mo>8<\/mn><\/mrow><\/math><\/div>\n\n\n\n

Divide by negative two and reverse the inequality sign.<\/p>\n\n\n\n

x<\/mi>\u2264<\/mo>\u2212<\/mo>4<\/mn><\/mrow><\/math><\/div>\n\n\n\n

For a compound inequality, perform the same operation on all three parts.<\/p>\n\n\n\n

\u2212<\/mo>3<\/mn><<\/mo>2<\/mn>x<\/mi>+<\/mo>1<\/mn>\u2264<\/mo>7<\/mn><\/mrow><\/math><\/div>\n\n\n\n
\u2212<\/mo>4<\/mn><<\/mo>2<\/mn>x<\/mi>\u2264<\/mo>6<\/mn><\/mrow><\/math><\/div>\n\n\n\n
\u2212<\/mo>2<\/mn><<\/mo>x<\/mi>\u2264<\/mo>3<\/mn><\/mrow><\/math><\/div>\n\n\n\n

Nature of roots<\/h2>\n\n\n\n

The nature of the roots describes the number and type of solutions of a quadratic equation. A quadratic can have:<\/p>\n\n\n\n