Inequalities are an officially tested skill in the Algebra domain of the Digital SAT, and this free, complete lesson covers everything in the College Board's Inequalities question category: reading, writing, and graphing basic inequalities, the properties of inequalities, multi-step inequalities and real-world word problems, and/or compound inequalities, absolute value inequalities and absolute value equations, tolerance and margin-of-error applications, and linear inequalities in two variables, including boundary lines and matching graphs to their equations. Each idea is followed by a fully worked example, and then a set of original practice problems with instant feedback, so you can build fluency across the entire topic in one place. Everything here is free, and you can keep practicing afterward with the full SAT Math Question Bank linked below.
An inequality is a mathematical sentence that compares two expressions using a symbol other than equals: >, <, ≥, or ≤. On the Digital SAT, you'll often need to translate a verbal phrase into an inequality, or read a number line graph and identify the inequality it represents.
| Verbal phrase | Inequality |
|---|---|
| less than, fewer than, below | < |
| greater than, more than, exceeds, above | > |
| at most, no greater than, no more than, less than or equal to | ≤ |
| at least, no less than, no fewer than, greater than or equal to | ≥ |
x > 2 – open circle, shaded right
x ≤ 2 – closed circle, shaded left
Worked Example: Write each statement as an inequality.
a. The number x is no greater than −2.
x ≤ −2
b. The amount of calories n meets or exceeds 1,200.
n ≥ 1,200
1. Which inequality represents "the number y is no more than 12"?
Explanation: "No more than" means the value can equal 12 or be smaller, so y ≤ 12.
2. Which inequality represents "the temperature t is at least 68 degrees"?
Explanation: "At least" means the value can equal 68 or be larger, so t ≥ 68.
3. On a number line graph of x > −2, what type of dot is used at −2, and which direction is shaded?
Explanation: Since −2 itself is not included (the symbol is >, not ≥), use an open circle. Since x must be greater than −2, shade to the right.
4. Which inequality is graphed by a closed dot at 5 with shading to the left?
Explanation: A closed dot means 5 is included, and shading left means values less than 5 are also included, so the inequality is x ≤ 5.
Inequalities follow rules similar to equations, with one critical exception. For all real numbers a, b, and c:
| Property | Rule |
|---|---|
| Addition and Subtraction | If a < b, then a + c < b + c and a − c < b − c |
| Multiplication and Division by a positive c | If a < b and c > 0, then ac < bc and ac < bc |
| Multiplication and Division by a negative c | If a < b and c < 0, then ac > bc and ac > bc |
That last rule is the one to memorize carefully: whenever you multiply or divide both sides of an inequality by a negative number, you must flip the inequality symbol. Adding or subtracting never flips the symbol, only multiplying or dividing by a negative does.
Worked Example: Solve −3x + 5 > 20.
−3x > 15 (Subtract 5 from each side)
x < −5 (Divide each side by −3, and flip the symbol)
1. Solve: −2x + 7 ≤ 19
Explanation: Subtract 7 from each side: −2x ≤ 12. Divide each side by −2, flipping the symbol: x ≥ −6.
2. Solve: 5 − x > 12
Explanation: Subtract 5 from each side: −x > 7. Multiply each side by −1, flipping the symbol: x < −7.
3. If a < b, which of the following is always true?
Explanation: Adding the same number to both sides never changes the direction of an inequality, so a + 5 < b + 5 is always true. Choice B reverses the inequality without a valid reason, C states the wrong direction, and D would mean a > b, which contradicts a < b.
4. If a < b and c is negative, which of the following is true?
Explanation: Multiplying both sides of a < b by a negative number c flips the inequality, giving ac > bc.
Solving a multi-step inequality follows the same process as solving a multi-step equation: distribute if needed, combine like terms, and isolate the variable using the properties above. The only new step to watch for is flipping the symbol if you multiply or divide by a negative number.
Worked Example: Solve 5x − 6 ≥ 3x + 8.
5x − 3x − 6 ≥ 8 (Subtract 3x from each side)
2x − 6 ≥ 8 (Simplify)
2x ≥ 14 (Add 6 to each side)
x ≥ 7 (Divide each side by 2; no flip needed since 2 is positive)
1. Solve: 7x + 4 < 3x + 24
Explanation: Subtract 3x from each side: 4x + 4 < 24. Subtract 4: 4x < 20. Divide by 4: x < 5.
2. Solve: 9 − 2x ≥ 4x − 15
Explanation: Add 2x to each side: 9 ≥ 6x − 15. Add 15: 24 ≥ 6x. Divide by 6: 4 ≥ x, which is the same as x ≤ 4.
3. Solve: 3(2x − 1) < 5x + 9
Explanation: Distribute: 6x − 3 < 5x + 9. Subtract 5x from each side: x − 3 < 9. Add 3: x < 12.
4. Solve: −4(x + 2) ≥ 2x − 4
Explanation: Distribute: −4x − 8 ≥ 2x − 4. Add 4x to each side: −8 ≥ 6x − 4. Add 4: −4 ≥ 6x. Divide by 6: −23 ≥ x, which is the same as x ≤ −23.
Many SAT inequality questions are wrapped inside a real-world scenario, often about budgets, fees, or limits. The key skill is translating the situation into an inequality using the same verbal phrases from the first section, then solving it just like any other inequality.
Worked Example: A moving truck rental costs $35 for the day plus $0.50 per mile driven. If Deja has budgeted no more than $95 for the rental, what is the greatest number of miles she can drive?
Let m be the number of miles. The total cost is 35 + 0.5m, and it must be no more than 95:
35 + 0.5m ≤ 95
0.5m ≤ 60 (Subtract 35 from each side)
m ≤ 120 (Divide each side by 0.5)
Deja can drive at most 120 miles.
1. A parking garage charges $5 for the first hour and $3 for each additional hour. If Maria has at most $23 to spend, what is the greatest number of additional hours she can park?
Explanation: Let h be the number of additional hours. The total cost is 5 + 3h, and it must be at most 23: 5 + 3h ≤ 23, so 3h ≤ 18, and h ≤ 6. She can park at most 6 additional hours.
2. A phone plan costs $30 per month plus $0.10 per text message beyond the included limit. Which inequality represents t, the number of extra text messages, that keeps the monthly bill under $50?
Explanation: The flat monthly fee, $30, plus $0.10 for each of the t extra texts, must total less than $50, giving 30 + 0.10t < 50.
3. A gym membership costs a $40 sign-up fee plus $25 per month. Kevin has budgeted no more than $215 total. What is the greatest number of months he can attend under this budget?
Explanation: Let m be the number of months. The total cost is 40 + 25m, and it must be no more than 215: 40 + 25m ≤ 215, so 25m ≤ 175, and m ≤ 7. He can attend for at most 7 months.
Two or more inequalities joined by the word and or or form a compound inequality. A compound inequality with "and" is true only if both individual inequalities are true, its graph is the intersection of the two solution sets. A compound inequality with "or" is true if at least one of the inequalities is true, its graph is the union of the two solution sets.
−6 < x < −3 ("and") – the segment between the two points
x ≤ 4 or x ≥ 5 ("or") – the two outer rays
Worked Example: Solve 9 − 3x > 18 and 5 + 2x > −7.
9 − 3x > 18 and 5 + 2x > −7
−3x > 9 and 2x > −12
x < −3 and x > −6 (dividing by −3 flips the first symbol)
Combined, the solution is −6 < x < −3.
1. Solve: x + 4 < 9 and x − 2 > −6
Explanation: x + 4 < 9 gives x < 5. x − 2 > −6 gives x > −4. Combined with "and," the solution is −4 < x < 5.
2. A number line graph shows open circles at −3 and 4, with the segment between them shaded. Which compound inequality does this represent?
Explanation: Open circles mean −3 and 4 are excluded, and shading between them means only values strictly between the two are included: −3 < x < 4.
3. Solve: 2x − 1 ≤ 7 or 3x + 5 > 20
Explanation: 2x − 1 ≤ 7 gives x ≤ 4. 3x + 5 > 20 gives x > 5. Combined with "or," any x satisfying either condition works: x ≤ 4 or x > 5.
4. A number x satisfies −5 < x ≤ 8. Which of the following is NOT a possible value of x?
Explanation: The strict inequality −5 < x means −5 itself is excluded. Since the other symbol is ≤, 8 is included. Only −5 is not a possible value.
5. The solution set of a compound inequality joined by "and" is called the ___ of the two individual solution sets.
Explanation: "And" requires both conditions to be true at once, which describes the intersection, the overlap, of the two solution sets. "Or" describes the union instead.
Since |x| represents the distance from x to zero on a number line, absolute value inequalities translate directly into compound inequalities. For any positive number c:
| Inequality | Equivalent compound inequality |
|---|---|
| |ax + b| < c | −c < ax + b < c |
| |ax + b| > c | ax + b < −c or ax + b > c |
In the table above, < can be replaced with ≤, and > can be replaced with ≥. The intuition: |x| < c means x is within c units of zero, so x is trapped between −c and c. And |x| > c means x is farther than c units from zero, so x is either far to the left or far to the right.
Worked Example: Solve |2x − 3| ≤ 9.
−9 ≤ 2x − 3 ≤ 9
−6 ≤ 2x ≤ 12 (Add 3 to each expression)
−3 ≤ x ≤ 6 (Divide each expression by 2)
1. Solve |x| < 7.
Explanation: |x| < 7 means x is within 7 units of zero, so −7 < x < 7.
2. Solve |x| > 5.
Explanation: |x| > 5 means x is farther than 5 units from zero, so x < −5 or x > 5.
3. Solve |3x + 6| ≤ 15.
Explanation: Rewrite as −15 ≤ 3x + 6 ≤ 15. Subtract 6 from each expression: −21 ≤ 3x ≤ 9. Divide each expression by 3: −7 ≤ x ≤ 3.
4. Solve |4x − 5| > 11.
Explanation: Rewrite as 4x − 5 < −11 or 4x − 5 > 11. First case: 4x < −6, so x < −32. Second case: 4x > 16, so x > 4.
5. For which value of c does |x + 4| < c have no solution?
Explanation: Since |x + 4| is never negative, the inequality |x + 4| < c has no solution whenever c is zero or negative. Only c = −2 fits that condition.
One of the most common real-world uses of absolute value inequalities is describing how far a measured value is allowed to differ from a target value, often called a tolerance or margin of error. If a quantity x must stay within a tolerance of t units from a target value v, that's written |x − v| ≤ t.
Worked Example: A recipe calls for 200 grams of flour, and a scale reading r is considered acceptable if it's within 8 grams of the target. Write an inequality for the acceptable readings, and find the resulting range.
|r − 200| ≤ 8
−8 ≤ r − 200 ≤ 8
192 ≤ r ≤ 208
1. A machine fills bags of flour to a target weight of 50 grams. A bag is acceptable if its weight w differs from the target by no more than 0.5 grams. Which inequality represents the acceptable weights?
Explanation: The weight w must be within 0.5 grams of the target, 50, and "no more than" includes equality, so |w − 50| ≤ 0.5.
2. Using the inequality from the previous problem, what is the range of acceptable weights?
Explanation: |w − 50| ≤ 0.5 becomes −0.5 ≤ w − 50 ≤ 0.5. Add 50 to each expression: 49.5 ≤ w ≤ 50.5.
3. A thermostat keeps a room's temperature T within 3 degrees of 70°F. What is the resulting range of acceptable temperatures?
Explanation: The condition is |T − 70| ≤ 3, which becomes −3 ≤ T − 70 ≤ 3. Add 70 to each expression: 67 ≤ T ≤ 73.
4. The inequality |x − 6| ≤ 4 is equivalent to which compound inequality?
Explanation: Rewrite as −4 ≤ x − 6 ≤ 4. Add 6 to each expression: 2 ≤ x ≤ 10.
The absolute value of a number is its distance from zero on a number line. Distance is never negative, so absolute value is always zero or positive, no matter what sign the original number had. The absolute value of a number n is written |n|.
|−8| = 8 and |8| = 8, since both are 8 units from zero.
Worked Example: The distance between −3 and the origin is 3, so |−3| = 3. The distance between 3 and the origin is also 3, so |3| = 3. Therefore, if |x| = 3, then x = 3 or x = −3.
1. What is |−9|?
Explanation: −9 is 9 units from zero on the number line, so |−9| = 9.
2. What is the distance between −12 and 0 on the number line?
Explanation: Distance is always nonnegative, so the distance between −12 and 0 is 12, which matches |−12| = 12.
3. If |x| = 15, what are the possible values of x?
Explanation: Both 15 and −15 are exactly 15 units from zero, so both values make the equation true.
4. Which value of x makes |x − 4| equal to 0?
Explanation: The only number with an absolute value of 0 is 0 itself, so x − 4 = 0, which means x = 4. This is the only solution, since a quantity is 0 units from zero only when it equals exactly zero.
To solve an absolute value equation like |expression| = a positive number, remember that whatever is inside the bars could have been positive or negative before the absolute value was applied. So you split the equation into two cases: one where the expression equals the positive value, and one where it equals the negative value, then solve each case separately.
Worked Example: Solve |4x − 3| = 13.
4x − 3 = 13 or 4x − 3 = −13
4x = 16 or 4x = −10
x = 4 or x = −52
1. Solve |2x + 1| = 9.
Explanation: Split into two cases: 2x + 1 = 9 gives 2x = 8, so x = 4. And 2x + 1 = −9 gives 2x = −10, so x = −5.
2. Solve |5x − 2| = 18.
Explanation: Split into two cases: 5x − 2 = 18 gives 5x = 20, so x = 4. And 5x − 2 = −18 gives 5x = −16, so x = −165.
3. Solve |x + 7| = 0.
Explanation: Since |0| = 0 is the only case, there's only one equation to solve, not two: x + 7 = 0, so x = −7. When an absolute value expression equals 0, there's exactly one solution instead of two.
4. Solve 2|3x − 1| = 16.
Explanation: First divide both sides by 2: |3x − 1| = 8. Then split into two cases: 3x − 1 = 8 gives 3x = 9, so x = 3. And 3x − 1 = −8 gives 3x = −7, so x = −73.
Since absolute value can never produce a negative result, an equation like |expression| = a negative number has no solution at all, no matter what the expression is. You can recognize these instantly without doing any algebra: just check the sign of the number on the other side of the equal sign.
Worked Example: Solve |x − 6| = −4.
Since −4 is negative, and an absolute value can never equal a negative number, this equation has no solution. There's no need to attempt splitting it into cases.
1. Solve |2x + 5| = −3.
Explanation: The right side, −3, is negative. Since absolute value is never negative, this equation has no solution.
2. Which of the following equations has no solution?
Explanation: Only |x + 1| = −6 sets an absolute value expression equal to a negative number, so it's the only equation with no solution.
3. Which value of k guarantees that |x + 3| = k has no solution?
Explanation: The equation has no solution whenever k is negative. Only k = −2 is negative; k = 0 gives exactly one solution, and k = 5 or k = 10 each give two solutions.
4. How many solutions does |x| = −7 have?
Explanation: No value of x has an absolute value of −7, since absolute value is always zero or positive. This equation has 0 solutions.
The absolute value function is written f(x) = |x|. Its graph is a V-shape: it comes down from the upper left, touches its lowest point at the origin, and rises back up to the upper right. A more general absolute value function, y = a|x − h| + k, has its vertex, the sharp point of the V, at (h, k). If a > 0, the graph opens upward and the vertex is the minimum point; if a < 0, the graph opens downward and the vertex is the maximum point.
The graph of f(x) = |x|, a V-shape with vertex at the origin.
Worked Example: What is the vertex of f(x) = 2|x − 3| + 5, and does it open upward or downward?
Comparing to y = a|x − h| + k, h = 3 and k = 5, so the vertex is (3, 5). Since a = 2 is positive, the graph opens upward and the vertex is the minimum point.
1. What is the vertex of the graph of f(x) = |x + 4| − 2?
Explanation: Rewrite as |x − (−4)| + (−2) to match y = a|x − h| + k. This gives h = −4 and k = −2, so the vertex is (−4, −2).
2. What is the vertex of g(x) = −3|x − 1| + 6, and does the graph open upward or downward?
Explanation: The vertex is (1, 6), read directly from h = 1 and k = 6. Since a = −3 is negative, the graph opens downward.
3. What is the value of f(x) = |x − 5| when x = 2?
Explanation: Substitute x = 2: f(2) = |2 − 5| = |−3| = 3.
4. For the graph of y = a|x| + k, if a > 0, the vertex is the graph's...
Explanation: When a > 0, the graph opens upward like a V, so the vertex is the lowest point on the graph, the minimum.
A linear inequality in two variables can be written in one of the forms ax + by < c, ax + by ≤ c, ax + by > c, or ax + by ≥ c. An ordered pair (a, b) is a solution to the inequality if the inequality is true when a and b are substituted for x and y. Unlike a linear equation, which has infinitely many solutions forming a line, a linear inequality has infinitely many solutions filling an entire half of the coordinate plane.
Worked Example: Is (3, −1) a solution to 2x − y ≤ 5?
2(3) − (−1) = 6 + 1 = 7
Is 7 ≤ 5? No, so (3, −1) is not a solution.
1. Is (2, 4) a solution to y < 3x − 1?
Explanation: Substitute: 3(2) − 1 = 5. Is 4 < 5? Yes, so (2, 4) is a solution.
2. Is (−1, 2) a solution to 2x + y ≥ 0?
Explanation: Substitute: 2(−1) + 2 = −2 + 2 = 0. Is 0 ≥ 0? Yes, this is true (the symbol includes equality), so (−1, 2) is a solution.
3. Which of the following points is NOT a solution to y ≤ −2x + 5?
Explanation: For (1, 4): −2(1) + 5 = 3. Is 4 ≤ 3? No, so this point fails. Checking the others: (0,0) gives 0 ≤ 5, true; (2,0) gives 0 ≤ 1, true; (−1,6) gives 6 ≤ 7, true.
4. For the inequality 3x − 2y > 6, which point is a solution?
Explanation: For (4, 0): 3(4) − 2(0) = 12. Is 12 > 6? Yes. Checking the others: (0,0) gives 0 > 6, false; (1,1) gives 1 > 6, false; (0,3) gives −6 > 6, false.
The equation you get by replacing the inequality symbol with an equal sign is called the boundary line. It divides the coordinate plane into two half-planes. If the inequality symbol is strict (< or >), the boundary line is dashed, since points on the line itself aren't included. If the symbol includes equality (≤ or ≥), the boundary line is solid.
To figure out which half-plane is the solution set, use the test-point method: pick any point not on the boundary line, usually the origin (0, 0) since it's easy to compute with, and substitute it into the original inequality. If the result is true, shade the half-plane containing that point. If false, shade the other half-plane. If the boundary line passes through the origin, pick a different test point instead.
Dashed line, shaded above – a strict inequality like y > mx + b
Solid line, shaded below – a non-strict inequality like y ≤ mx + b
Worked Example: Determine whether the boundary line for 2y + x ≤ 4 is solid or dashed, and which side of the line contains the shaded solution set.
Since the symbol is ≤, the boundary line is solid.
Testing the origin: 2(0) + 0 = 0. Is 0 ≤ 4? Yes, so the shaded region is the side of the line containing the origin.
1. Is the boundary line for y > 3x − 2 solid or dashed?
Explanation: The symbol > is strict, meaning points on the line itself are not solutions, so the boundary line is dashed.
2. For the inequality 4x − y ≥ 8, should the side containing the origin be shaded?
Explanation: Testing the origin: 4(0) − 0 = 0. Is 0 ≥ 8? No, so the origin is not a solution, and the side of the line containing the origin should not be shaded.
3. Is (0, 0) a valid test point for 3x − 2y > 2, and what does testing it tell you?
Explanation: The boundary line 3x − 2y = 2 does not pass through (0, 0), since 3(0) − 2(0) = 0 ≠ 2, so (0, 0) is a valid test point. Substituting gives 0 > 2, which is false, so the shaded region is the side of the line that does not contain the origin.
4. Why can't (0, 0) be used as a test point for the inequality y > 2x?
Explanation: The boundary line y = 2x passes directly through the origin, since 2(0) = 0. A test point can't lie on the boundary line itself, so a different point, like (1, 0), should be used instead.
5. What is the boundary line equation for 2y + x ≤ 4, written in slope-intercept form?
Explanation: Solve for y: 2y = 4 − x, so y = −12x + 2.
On the Digital SAT, this skill is tested by describing or showing a graph and asking which inequality matches it, or the reverse: giving an inequality and asking which graph matches. Reading a graph means noting three things: the slope and y-intercept of the boundary line, whether the line is solid or dashed, and which side is shaded.
Worked Example: A graph shows a dashed line passing through (0, −3) and (1, −1), with the region above the line shaded. Which inequality does this graph represent?
Find the slope: m = −1 − (−3)1 − 0 = 2. The y-intercept is −3, so the boundary line is y = 2x − 3.
Since the line is dashed, the symbol is strict. Since the shading is above the line, the symbol is >.
y > 2x − 3
1. A graph shows a dashed line passing through (0, −3) and (1, −1), with the region above the line shaded. Which inequality does this graph represent?
Explanation: This is the worked example above: slope 2, y-intercept −3, dashed line (strict), shaded above (>), giving y > 2x − 3.
2. A graph shows a solid line passing through (4, 0) and (0, 4), with the region containing the origin shaded. Which inequality does this graph represent?
Explanation: The line through (4, 0) and (0, 4) has slope −1 and equation x + y = 4. The line is solid, so the symbol includes equality. Testing the origin: 0 + 0 = 0, and since the origin is shaded, 0 must satisfy the inequality, so the symbol is ≤, giving x + y ≤ 4.
3. Which description matches the graph of y ≤ −3x + 6?
Explanation: The symbol ≤ includes equality, so the line is solid. The line has slope −3 and y-intercept 6. Since y is less than or equal to the line's expression, the shading is below the line.
4. The point (2, 5) is tested in the inequality y > 4x − 1. Is (2, 5) a solution?
Explanation: Substitute: 4(2) − 1 = 7. Is 5 > 7? No, so (2, 5) is not a solution.
5. A graph shows a dashed vertical line at x = 3, with shading to the right of the line. Which inequality does this graph represent?
Explanation: A dashed line means the symbol is strict, and shading to the right means x-values greater than 3 are included, so the inequality is x > 3.
Yes. "Linear inequalities in 1 or 2 variables" is one of the officially listed question types in the Algebra domain of the Digital SAT Math section, alongside linear equations, linear functions, and systems of equations. Compound and absolute value inequalities aren't singled out as a separate named category and appear less often, usually through a tolerance word problem.
Think of a number line: 2 < 5 is true. Multiply both sides by −1, and you get −2 and −5. But −2 is actually greater than −5, since it's farther to the right. So the inequality has to flip, −2 > −5, to stay true.
|x| < c describes values within a fixed distance of zero, a bounded range that needs "and" (or, equivalently, a three-part inequality). |x| > c describes values farther than a fixed distance from zero, two separate unbounded pieces that need "or."
Because two different numbers, one positive and one negative, can have the same absolute value. For example, both 5 and −5 have an absolute value of 5, so any equation that simplifies to |something| = 5 needs to account for both possibilities.
Yes. Look at what the absolute value expression is set equal to. If it's positive, expect two solutions. If it's exactly 0, expect one solution. If it's negative, expect no solution. Checking this first can save you from doing unnecessary algebra.
A linear function graphs as a single straight line, while the absolute value function is really two linear pieces joined at the vertex, one with a positive slope and one with a negative slope. That's why its graph is a V-shape instead of a single straight line.
No. Since the Digital SAT never asks you to draw a graph, the more valuable skill is reading a graph that's already shown to you, or reasoning about an inequality algebraically using test points, exactly what this lesson focuses on.
If the inequality is already solved for y, like y > mx + b, the symbol tells you directly: > or ≥ means shade above the line, and < or ≤ means shade below the line. The test-point method is most useful as a double-check, or when the inequality is still in standard form.
A linear equation's solutions form a single line, every point exactly on that line. A linear inequality's solutions fill an entire half of the plane, either every point on one side of the line, or every point on that side plus the line itself.
The Inequalities quizzes in the SAT Math Question Bank include additional original problems on this topic, along with quizzes covering every other Algebra, Advanced Math, Problem-Solving and Data Analysis, and Geometry and Trigonometry skill on the Digital SAT.
