Praxis Core Algebra Summary

1. Add 7 to both sides: 4x = 28.
2. Divide both sides by 4: x = 7.

Check: 4 × 7 − 7 = 28 − 7 = 21.

Wrong choices

• "x = 3.5": This subtracts 7 from 21 instead of adding: 4x = 14. To undo − 7, add 7 to both sides.
• "x = 28": This adds 7 to get 4x = 28 but forgets to divide by 4.
• "x = 5.25": This divides 21 by 4 without first moving the 7.
• "x = 112": This multiplies 28 by 4 instead of dividing.

The $25 fee is paid once, and $15 is paid for each of the m months, so the total is 25 + 15m.

Check: for 2 months, 25 + 15 × 2 = $55, which is $25 + $15 + $15.

Wrong choices

• "15 + 25m": This switches the fees. The $15 is charged each month; the $25 is paid only once.
• "40m": This charges the sign-up fee every month.
• "15m": This leaves out the one-time $25 sign-up fee.
• "25 − 15m": The monthly fee adds to the cost; it is not subtracted.

1. Distribute: 3x − 12 = 2x + 5.
2. Subtract 2x from both sides: x − 12 = 5.
3. Add 12 to both sides: x = 17.

Check: 3(17 − 4) = 3 × 13 = 39, and 2 × 17 + 5 = 39.

Wrong choices

• "x = 9": This multiplies only x by 3 and leaves −4, giving 3x − 4 = 2x + 5. The 3 must multiply both terms.
• "x = −7": This subtracts 12 instead of adding it when moving it across: x = 5 − 12.
• "x = 3.4": This adds 2x to the left side (5x = 17). To move 2x, subtract it from both sides.
• "x = −17": This makes a sign error; checking −17 gives −63 on the left and −29 on the right.

1. a^2 = 4 × 4 = 16.
2. 3a^2 = 3 × 16 = 48.
3. 2b = 2 × 5 = 10.
4. 48 − 10 = 38.

Check: 3 × 16 = 48 and 48 − 10 = 38.

Wrong choices

• "43": This subtracts only b (5) instead of 2b (10).
• "58": This adds 2b instead of subtracting it.
• "14": This doubles a instead of squaring it: 3 × 4 × 2 = 24, and 24 − 10 = 14.
• "134": This squares 3a (12^2 = 144). Only a is squared, then multiplied by 3.

"Twice a number" is 2n. "Five less than" that is 2n − 5. "Is 17" means = 17, so the equation is 2n − 5 = 17.

Check: the solution n = 11 gives 2 × 11 − 5 = 17.

Wrong choices

• "5 − 2n = 17": "Five less than" something means subtract 5 FROM it, so the 5 comes second.
• "2(n − 5) = 17": This subtracts 5 from the number before doubling. The sentence doubles the number first.
• "2n + 5 = 17": "Less than" means subtract, not add.
• "5n − 2 = 17": This switches the roles of 5 and 2. "Twice" means multiply by 2.

1. Subtract 9 from both sides: x > 4 − 9.
2. x > −5.

Check: x = 0 (which is greater than −5) gives 0 + 9 = 9 > 4, true.

Wrong choices

• "x > 13": This adds 9 to both sides instead of subtracting 9.
• "x < −5": The sign flips only when you multiply or divide by a negative number. Subtracting 9 does not flip it.
• "x > 5": 4 − 9 is −5, not 5.
• "x < 13": This adds 9 and also flips the sign; both steps are wrong.

1. Subtract 4 from both sides: −3x ≤ 15.
2. Divide by −3 and flip the sign: x ≥ −5.

Check: x = 0 (which is ≥ −5) gives 4 ≤ 19, true; x = −6 gives 22 ≤ 19, false.

Wrong choices

• "x ≤ −5": This forgets to flip the sign when dividing by −3.
• "x ≥ 5": This drops the negative sign: 15 ÷ (−3) = −5, not 5.
• "x ≤ 5": This loses the negative sign and also forgets to flip the inequality.
• "x ≥ −23/3": This adds 4 to 19 instead of subtracting it.

1. 30 + 0.10t ≤ 45.
2. Subtract 30: 0.10t ≤ 15.
3. Divide by 0.10: t ≤ 150.

Check: 150 texts cost $15, and $30 + $15 = $45, exactly the limit.

Wrong choices

• "450": This divides the whole $45 by $0.10 and forgets the $30 monthly charge.
• "300": This divides the $30 monthly charge by $0.10; that has nothing to do with the texts allowed.
• "15": $15 is the money left for texts; it must be divided by $0.10 per text.
• "750": This adds $30 and $45 and divides by $0.10.

1. Let a = adult tickets, so child tickets = 50 − a.
2. Money: 12a + 7(50 − a) = 500 → 12a + 350 − 7a = 500.
3. 5a = 150, so a = 30.

Check: 30 × $12 = $360 and 20 × $7 = $140; $360 + $140 = $500.

Wrong choices

• "20": 20 is the number of child tickets, not adult tickets.
• "25": This assumes half of the tickets were for adults. That would give only $475.
• "42": This divides $500 by $12, as if every ticket were an adult ticket.
• "71": This divides $500 by $7, as if every ticket were a child ticket.

1. Change in y: 11 − 3 = 8.
2. Change in x: 6 − 2 = 4.
3. Slope: 8 ÷ 4 = 2.

Check: starting at (2, 3), going 4 right and 8 up reaches (6, 11).

Wrong choices

• "1/2": This divides the change in x by the change in y. Slope is rise (y) over run (x).
• "8": 8 is the rise (11 − 3). It must be divided by the run (6 − 2 = 4).
• "3.5": This adds the y-values (11 + 3) instead of subtracting them.
• "−2": Both y and x increase from one point to the other, so the slope is positive.