SAT linear equations and inequalities: review and practice

• Solve a one-variable linear equation, then find the value of an expression like 2x − 1.
• Decide whether an equation has exactly one solution, no solution or infinitely many.
• Turn a word problem (a rental fee, an hourly rate, a budget) into an equation.
• Solve inequalities, including flipping the sign when you divide by a negative number.
• Find values that meet several conditions at once, such as the most whole batches you can make with what's left.

• Answering for x when the question asks for an expression. If 3(x + 4) = 27, then x + 4 = 9, so x + 1 = 6. Stopping at x = 5 lands on a trap choice.
• Forgetting to flip the inequality. −2x < 10 gives x > −5, not x < −5. Flip the sign whenever you multiply or divide both sides by a negative number.
• Distributing to only the first term. −3(x − 4) is −3x + 12, not −3x − 4 or −3x − 12.
• Charging a flat fee twice. If parking costs $50 for the first 2 hours and $15 for each extra hour, the cost for h hours is 50 + 15(h − 2), not 50 + 15h.
• Checking one condition and ignoring the rest. In a word problem with a budget and a weight limit, the largest number that fits the budget may break the weight limit. Test your answer against every constraint.
• Rounding up a whole-number answer. If the math says 3.7 batches fit, the answer is 3; you can't make 0.7 of a batch with the ingredients you have.

1. Question 1 · Easy

   If 7(2x − 5) = 84, what is the value of 2x − 1?

   • A8
   • B8.5
   • C12
   • D16
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   D. Dividing by 7 gives 2x − 5 = 12. Since 2x − 1 = (2x − 5) + 4, the requested value is 16; choice A subtracts this 4, choice B reports x, and choice C reports the original group.

2. Question 2 · Easy

   How many real solutions does the equation 11x − 8 = −8 + 11x have?

   • AExactly two real solutions
   • BNo real solutions
   • CExactly one real solution
   • DInfinitely many real solutions
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   D. Reordering the terms on the right gives 11x − 8, so every real value of x satisfies the equation. Therefore, there are infinitely many real solutions. Choice B mistakes cancellation for a contradiction, choice C restricts the solutions to zero, and choice A substitutes a finite count for all real numbers.

3. Question 3 · Medium

   An outdoor retailer rents a tent for $142 for the first 3 days and $24 for each additional day. The total rental cost is c dollars for d days, where d is an integer and 3 ≤ d ≤ 12. Which equation represents the total rental cost?

   • Ac = 24d + 118
   • Bc = 142d − 284
   • Cc = 24d + 70
   • Dc = 24d + 142
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   C. The cost is 142 + 24(d − 3) = 24d + 70, which gives 142 dollars at d = 3 and increases by 24 dollars per additional day. The constants 142 and 118 double-charge all or part of the covered block, while 142d − 284 has the correct 3-day cost but an incorrect daily increase of 142 dollars.

4. Question 4 · Medium

   Which point (x, y) is a solution to the system of inequalities 2x − 3y < −6 and x + 2y ≤ 1?

   • A(0, 3)
   • B(0, 1)
   • C(−3, 1)
   • D(−3, 0)
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   C. The system is y > (2x + 6)/3 and y ≤ (1 − x)/2; at x = −3 this gives 0 < y ≤ 2, so (−3, 1) works. Choice D is on the excluded first boundary, choice A gives 6 > 1 in the second inequality, and choice B gives −3 < −6 in the first inequality, which is false.

5. Question 5 · Medium

   A soap maker has 19.5 kilograms of shea butter. Each large batch of soap uses 2.25 kilograms of shea butter, and each small batch uses 0.8 kilogram. After making 6 large batches, what is the greatest number of whole small batches the soap maker can make with the remaining shea butter?

   • A2
   • B6
   • C7
   • D8
   Show the answer

   C. The large batches use 6(2.25) = 13.5 kilograms, leaving 6 kilograms, so the small-batch count is at most 6/0.8 = 7.5 and the maximum whole count is 7. Seven small batches use 5.6 of the remaining kilograms, while eight use 6.4; choice A uses the large-batch rate, choice B reports remaining kilograms, and choice D exceeds the supply.

6. Question 6 · Hard

   The equation a(4 − 3x) + b(3x − 4) = 5a − 2b holds, where a and b are constants with a > b > 0. What is the value of (6x − 8)/(5a − 2b)?

   • A2/(a − b)
   • B−2/(a + b)
   • C−2/(a − b)
   • D−1/(a − b)
   Show the answer

   C. Since 3x − 4 = −(4 − 3x), the equation gives (a − b)(4 − 3x) = 5a − 2b, with a − b > 0 and 5a − 2b > 0. Using 6x − 8 = −2(4 − 3x), the requested ratio is −2/(a − b); choice B adds the coefficients, choice D loses the factor 2, and choice A loses the negative sign.

Type the equation exactly as given, such as 3(x + 4) = 27, and Desmos draws a vertical line at the solution; click it to read x. Then type the expression the question asks for with that number in place of x. For a system of inequalities, enter both inequalities and see which answer choice's point lands where the two shadings overlap. A point on a dashed boundary line doesn't count.

Ashore's SAT course starts with a 15-minute check that finds the question types costing you the most points, then gives you lessons and practice for each.

• Linear Functions and Systems
• Ratios, Rates, Percentages, and Units
• One- and Two-Variable Data
• Probability and Statistical Inference
• Equivalent Expressions and Quadratics
• Nonlinear Functions and Exponentials
• Lines, Angles, and Triangles
• Right Triangles and Trigonometry
• Circles, Area, and Volume
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