Algebra expressions practice: 6 questions with answers

Algebra expression questions come in two shapes. Some give you words and ask for the expression. Some give you an expression or equation and ask you to simplify or solve. The first three steps below are the same for both. After that the two shapes part ways: an expression has no two sides, so you combine like terms where they stand, while an equation lets you move terms across the equals sign and usually ends with a division.

Six steps

1. Name the unknown. Write “let ( x ) = the number of notebooks” or whatever the question is about. One letter, one meaning.
2. Translate one phrase at a time. Work left to right through the sentence and write each piece in symbols before you join them.
3. Clear parentheses. Multiply the number outside by every term inside, and carry its sign.
4. Clear fractions, if it is an equation. Multiply every term on both sides by the smallest number all the denominators divide into. An expression has no second side to balance, so leave its fractions alone and write them over a common denominator instead.
5. Finish it.
   • Expression: combine like terms where they are. Add the ( x ) terms together, add the plain numbers together, and stop. Nothing crosses an equals sign, because there is no equals sign. Your answer is a shorter expression.
   • Equation: put the variable terms on one side and the plain numbers on the other, then divide both sides by the number in front of the variable. Your answer is a value.
6. Check it.
   • Expression: pick a sample value, put it into the original and into your answer, and see that both give the same number. A single matching sample does not prove the two expressions are equal for every value — it catches sign slips and dropped terms. Try a second value, such as a negative one, if you are unsure.
   • Equation: put your solution back into the original equation and work out each side. If both sides come out equal, the solution is correct.

Word order traps

Some phrases reverse when you write them in symbols.

Read “less than” and “subtracted from” as instructions to flip the order. “The sum of” needs parentheses when another operation applies to the whole sum, as in “twice the sum of ( x ) and 4”, which is ( 2(x + 4) ). When nothing else acts on the sum, no parentheses are needed: “the sum of ( x ) and 4” is just ( x + 4 ).

Step 3 in practice

Simplify ( 5 - 3(2x - 1) ).

The ( -3 ) multiplies both terms inside: ( -3 \cdot 2x = -6x ) and ( -3 \cdot (-1) = +3 ). So the expression becomes ( 5 - 6x + 3 ), which is ( 8 - 6x ).

This is an expression, so check it with a sample value. Try ( x = 2 ). Original: ( 5 - 3(4 - 1) = 5 - 9 = -4 ). Answer: ( 8 - 12 = -4 ). Same. Try ( x = 0 ) as well. Original: ( 5 - 3(0 - 1) = 5 + 3 = 8 ). Answer: ( 8 - 0 = 8 ). Same again. Two agreeing samples are not a proof, but they would have caught the usual mistake here, which is dropping the minus sign and writing ( 5 - 6x - 3 ).

Step 4 in practice

Solve ( \frac{1}{2}x + 1 = \frac{1}{5}x + 4 ).

Both denominators divide into 10, so multiply every term by 10: ( 5x + 10 = 2x + 40 ). Subtract ( 2x ): ( 3x + 10 = 40 ). Subtract 10: ( 3x = 30 ). Divide by 3: ( x = 10 ).

This is an equation, so check it by putting the solution back in. With ( x = 10 ): the left side is ( \frac{1}{2}(10) + 1 = 6 ) and the right side is ( \frac{1}{5}(10) + 4 = 6 ). The sides match, so ( x = 10 ) is correct.

One extra rule for inequalities

Everything above applies, with one addition: when you multiply or divide both sides by a negative number, the inequality sign turns around.

Solve ( 4 - 2x < 10 ). Subtract 4: ( -2x < 6 ). Divide by ( -2 ) and flip the sign: ( x > -3 ).

Check a value that should work, ( x = 0 ): ( 4 - 0 = 4 < 10 ), true. Check one that should not, ( x = -4 ): ( 4 + 8 = 12 < 10 ), false. The answer holds.

The questions start easy and get harder. Try each one before you look at the explanation.