PSAT 8/9 Math · Lesson 2 of 4
Systems and inequalities
Use two conditions at the same time, and keep the direction of an inequality correct when you multiply or divide by a negative number.
Two statements, one solution
A system asks for values that satisfy every equation together. If x + y = 10 and x − y = 4, adding the equations cancels y and gives 2x = 14, so x = 7 and y = 3. A solution that fits only the first equation is not finished.
Substitution is useful when one equation already isolates a variable. Replace that variable in the other equation, solve, then return to find the second value. Check both original equations, not only the equation you solved last.
The direction of an inequality
An inequality uses the same balancing steps as an equation, with one difference: multiplying or dividing by a negative number reverses the symbol. So −2x < 8 becomes x > −4, not x < −4.
On a number line or graph, decide whether the boundary is included. “At least 12” includes 12 and uses ≥. “Fewer than 12” does not include 12 and uses <.
Worked example
Two numbers add to 18. The first is 6 more than the second. What are the numbers?
- Let the numbers be a and b. Then a + b = 18 and a = b + 6.
- Substitute: (b + 6) + b = 18, so 2b = 12 and b = 6.
- Then a = 12. Check: 12 + 6 = 18 and 12 is 6 more than 6.
Why this works. Write the comparison and the total as separate equations before substituting.
Check your understanding
- State what each variable counts before writing a system.
- Check a proposed pair in every equation.
- Reverse the inequality when you divide by a negative number, and say whether the endpoint is included.