Why it matters on the HSPT
Algebra on the HSPT is a small, well-defined set: turn a sentence into an expression, plug a number into an expression, solve for x in one or two steps, and handle a simple inequality. Those skills appear in the Mathematics section directly and in Quantitative Skills as comparisons of expressions in x. Students who have started Algebra I sometimes overthink these; the test wants the short version.
The rules
Words to symbols. A number and a variable next to each other multiply: 3x is 3 times x. Sum means add, difference means subtract, product means multiply, quotient means divide. Twice a number is 2x. Five more than a number is x plus 5. Five less than a number is x minus 5, and the order matters: less than reverses the words. A number decreased by 4 is x minus 4. Half of a number is x over 2.
Evaluating. Replace the variable with the number, in parentheses, and follow the order of operations. If x is 3, then 2x squared plus 1 is 2 times 9 plus 1, which is 19, not 2 times 3 squared as 36 plus 1. Put the number in parentheses every time so the exponent lands on the right thing.
Combining like terms. Terms with the same variable add: 3x plus 5x is 8x. A number and a variable do not combine: 3x plus 5 stays 3x plus 5. The distributive property spreads a multiplier across parentheses: 2 times the quantity x plus 4 is 2x plus 8.
One-step equations. Undo whatever is being done to x by doing the opposite to both sides. If x plus 7 equals 12, subtract 7: x is 5. If 4x equals 28, divide by 4: x is 7. If x over 3 equals 6, multiply by 3: x is 18.
Two-step equations. Undo in reverse order: first the add or subtract, then the multiply or divide. For 5x minus 4 equals 26, add 4 to get 5x equals 30, then divide to get x equals 6. Check by substituting: 5 times 6 minus 4 is 26.
Inequalities. Solve exactly like an equation, with one exception: multiplying or dividing both sides by a negative flips the sign. Negative 2x is greater than 8 becomes x is less than negative 4. On a number line, a solid dot means the number is included, an open dot means it is not.
Patterns and rules. A table of inputs and outputs asks for the rule. Test the simple ones first: add a constant, multiply by a constant, then multiply and add. If 1 gives 5, 2 gives 8, and 3 gives 11, the outputs rise by 3 each time, so the rule multiplies by 3 and adds 2.
Worked examples
Example 1
Which expression means 7 less than twice a number?
- A.7 − 2n
- B.2n − 7
- C.2(n − 7)
- D.2n + 7
Show answer
Answer: B: 2n − 7
Twice a number is 2n. Seven less than that means subtract 7 from it: 2n minus 7. Choice A reverses the subtraction, which is the standard trap with less than.
Example 2
If x = 4, what is the value of 3x² − 2x?
- A.16
- B.40
- C.136
- D.144
Show answer
Answer: B: 40
Substitute 4 in parentheses. Three times 4 squared is 3 times 16, which is 48. Two times 4 is 8. Forty-eight minus 8 is 40. Choice C, 136, squares 3 times 4 as 12 squared, applying the exponent to the wrong thing.
Example 3
Solve: x + 9 = 4
- A.−13
- B.−5
- C.5
- D.13
Show answer
Answer: B: −5
Subtract 9 from both sides: x equals 4 minus 9, which is negative 5. Check: negative 5 plus 9 is 4. Choice D adds instead of subtracting.
Example 4
Solve: 4x − 6 = 18
- A.3
- B.4
- C.6
- D.24
Show answer
Answer: C: 6
Add 6 to both sides: 4x equals 24. Divide by 4: x equals 6. Check: 4 times 6 is 24, minus 6 is 18. Choice A, 3, comes from dividing 18 by 4 and rounding, skipping the first step.
Example 5
For which value of x is 2x + 3 greater than 11?
- A.2
- B.3
- C.4
- D.5
Show answer
Answer: D: 5
Subtract 3: 2x is greater than 8. Divide by 2: x is greater than 4. The only choice greater than 4 is 5. Four itself does not work, because 2 times 4 plus 3 is exactly 11, not more than 11.
Example 6
Simplify: 3(x + 2) + 4x
- A.7x + 2
- B.7x + 6
- C.12x
- D.3x + 6
Show answer
Answer: B: 7x + 6
Distribute the 3: 3x plus 6. Add the 4x: 7x plus 6. Choice A forgets to multiply the 2 by 3, and choice C wrongly combines the 6 with the x terms.
Example 7
A rule turns 2 into 7, 3 into 10, and 5 into 16. What does it turn 8 into?
- A.19
- B.24
- C.25
- D.27
Show answer
Answer: C: 25
The outputs rise by 3 for each increase of 1 in the input, so the rule multiplies by 3, and 3 times 2 is 6, one less than 7, so it then adds 1. Check: 3 times 5 plus 1 is 16. For 8, the rule gives 24 plus 1, which is 25. Choice B, 24, forgets the add-1 step.
Common mistakes
- Reversing less than. Five less than x is x minus 5, not 5 minus x.
- Squaring the coefficient. In 3x squared, only x is squared. Use parentheses when substituting.
- Undoing the steps in the wrong order. Add or subtract first, then multiply or divide.
- Forgetting to flip the inequality. Dividing by a negative reverses the sign.
- Combining unlike terms. 4x plus 3 is not 7x.
Practice
These skills are the whole of pre-algebra, appear in word problems that set up an equation, and drive non-geometric comparison items that compare expressions. The integers lesson covers the sign rules that solving depends on.