Solve linear equations and inequalities in one variable · lesson

Distribute, collect, solve

💡 Simplify each side, gather the x terms on one side and the numbers on the other, then divide.

First clear parentheses and combine like terms on each side.

Then move the x terms to one side and the constants to the other. Divide by the coefficient of x.

Check by substituting your answer into the original equation.

🧠 Let's work it out

Solve 3(x − 2) + 5 = x + 9.

What do you think comes next?

💡 Multiply every term on both sides by a common denominator (or by 10) so the equation has only whole numbers.

Fractions are easier to remove than to work with. Find a number every denominator divides into, and multiply every term by it.

Put parentheses around each numerator first, so a minus sign in front of a fraction reaches every term on top.

For decimals, multiply every term by 10 or 100.

🧠 Let's work it out

Solve (x + 3)/2 = (x + 9)/3.

Multiply both sides by 6: 3(x + 3) = 2(x + 9).
Distribute: 3x + 9 = 2x + 18.
Subtract 2x and 9: x = 9.
Decimals clear the same way, times 10: 0.3(x − 2) = 0.2x + 1 becomes 3(x − 2) = 2x + 10, so x = 16.
An inequality clears the same way: multiplying by a positive number keeps the sign.

What do you think comes next?

denominator: the bottom of a fraction
least common denominator: the smallest number every denominator divides into

💡 Solve an inequality with the same steps as an equation; adding, subtracting, or dividing by a positive keeps the sign.

An inequality's solution is a set of numbers, like x ≤ 12. Use the same steps as for an equation.

Adding or subtracting anything keeps the sign. Multiplying or dividing by a positive number keeps it too.

Type ≤ as <= and ≥ as >=.

🧠 Let's work it out

Solve (3/4)x + 2 ≤ 11.

What do you think comes next?

💡 Multiplying or dividing both sides by a negative number reverses the inequality.

Multiplying by a negative reverses order: 3 < 7, but −3 > −7. So the sign must flip.

To avoid this, you can move the x terms so the coefficient ends up positive.

Always test one number from your answer in the original inequality.

🧠 Let's work it out

Solve −3x + 5 > x − 11.

What do you think comes next?

💡 To find where two plans cost the same, set their cost expressions equal and solve.

Two plans often have a different flat fee and a different rate.

Write each cost as fee + rate × amount. Set them equal and solve.

🧠 Let's work it out

Gym A costs $30 plus $5 per visit. Gym B costs $50 plus $3 per visit. After how many visits do they cost the same?

What do you think comes next?