Linear inequalities
Linear inequalities
Learn to solve, graph, interpret, and combine inequalities — from a single number-line condition to overlapping regions on the coordinate plane.
The big idea
An equation usually describes an exact value or a line. An inequality describes a set of possible values or a region of possible points.
One-variable example: \(x=4\)
Two-variable example: \(y=2x+1\)
A precise value or a line.
One-variable example: \(x>4\)
Two-variable example: \(y>2x+1\)
A set of values or a region.
1. Read the inequality before solving
Words such as at least, at most, more than, and less than tell you which comparison symbol to use.
| Words | Symbol | Meaning |
|---|---|---|
| at least | \(\ge\) | greater than or equal to |
| at most | \(\le\) | less than or equal to |
| more than | \(>\) | strictly greater than |
| less than | \(<\) | strictly less than |
2. One-step inequalities
Solve them like equations — with one critical exception: multiplying or dividing by a negative number reverses the inequality.
Add or subtract the same number
Multiply or divide by a positive number
3. The critical rule: a negative reverses the sign
When you multiply or divide both sides of an inequality by a negative number, the order of the numbers reverses.
4. Negative numbers can make inequalities feel unfamiliar
On a number line, numbers farther to the right are greater. Therefore \(-2>-5\), even though 5 is greater than 2 in absolute value.
When checking a solution, compare the actual numbers — not their distances from zero.
5. Multi-step inequalities
Use the same algebraic structure you used for equations. Keep the inequality balanced while isolating the variable.
6. Compound inequalities
A compound inequality can describe values between two boundaries.
This means \(x\) is at least \(-2\) but less than 5. The left endpoint is included; the right endpoint is not.
7. Inequalities from real-world constraints
SAT questions often ask you to turn a practical condition into an inequality before solving it.
“No more than 48 students” → \(n\le48\)
“At least 750 points” → \(p\ge750\)
“Cannot spend more than $320” → \(C\le320\)
“Must be farther than 12 km” → \(d>12\)
8. Inequalities in two variables
A linear equation such as \(y=2x-1\) gives a boundary line. The inequality \(y>2x-1\) gives the points above that line.
Solid or dashed?
Use a solid boundary because points on the line are included.
Use a dashed boundary because points on the line are excluded.
9. How to decide which side to shade
When the inequality is not already solved for \(y\), or when the graph is unfamiliar, use a test point.
10. Which ordered pair satisfies an inequality?
Do not try to judge the graph by appearance alone. Substitute the ordered pair into the inequality.
11. Systems of linear inequalities
A system requires all inequalities to be true at the same time. On a graph, that means we first identify each solution region, then keep only the part where the regions overlap.
12. Finding the boundary intersection
When two boundary lines meet, their intersection can help describe the corner of the solution region. Solve the two boundary equations as a system.
The boundary lines meet at \((2,4)\). Whether that point belongs to the final solution depends on whether the original inequalities include their boundaries.
13. Integer constraints can change the final answer
Some SAT contexts describe a count of people, items, days, or other quantities that must be whole numbers.
Here 11 is already an integer, so it is the smallest possible value. If the algebra produced \(n\ge11.4\) and \(n\) represented a number of students, the smallest possible value would be 12.
14. A reliable SAT inequality strategy
Practice: linear inequalities
These questions are original SATMath800 practice. They use the same kinds of reasoning emphasized in College Board Algebra materials without reproducing source questions.
What is the solution to \(5x-8\ge27\)?
- A) \(x\ge5\)
- B) \(x\ge7\)
- C) \(x\le7\)
- D) \(x\le5\)
Which inequality is equivalent to \(-6x+4<22\)?
- A) \(x<-3\)
- B) \(x>3\)
- C) \(x<- \frac{13}{3}\)
- D) \(x>\frac{13}{3}\)
Which statement is true?
- A) \(-8>-3\)
- B) \(-5<-9\)
- C) \(-2>-7\)
- D) \(0<-4\)
A school can purchase no more than 36 tablets. If \(t\) represents the number of tablets purchased, which inequality represents the condition?
- A) \(t<36\)
- B) \(t\le36\)
- C) \(t>36\)
- D) \(t\ge36\)
A machine produces 18 parts each hour. After \(h\) hours, at least 144 parts must have been produced. What is the minimum whole-number value of \(h\)?
- A) 6
- B) 7
- C) 8
- D) 9
Which value satisfies the compound inequality below?
- A) \(-4\)
- B) \(-5\)
- C) \(3\)
- D) \(4\)
Which inequality is equivalent to \(2-4x\le18\)?
- A) \(x\le-4\)
- B) \(x\ge-4\)
- C) \(x\le-5\)
- D) \(x\ge-5\)
Which ordered pair satisfies \(2x+y<9\)?
- A) \((4,2)\)
- B) \((3,4)\)
- C) \((2,4)\)
- D) \((5,0)\)
Which boundary should be used when graphing \(y\ge-3x+5\)?
- A) A dashed line, because the line is excluded
- B) A solid line, because the line is included
- C) A dashed line, because the slope is negative
- D) A solid line, because the slope is negative
Which point lies in the solution region of \(y< x+3\)?
- A) \((0,4)\)
- B) \((2,6)\)
- C) \((1,5)\)
- D) \((3,5)\)
Which ordered pair satisfies both \(y\ge x-1\) and \(y<5\)?
- A) \((2,0)\)
- B) \((4,2)\)
- C) \((1,1)\)
- D) \((6,4)\)
A club has \(x\) adult members and \(y\) student members. The club can have no more than 80 members. Which inequality represents this condition?
- A) \(x+y<80\)
- B) \(x+y>80\)
- C) \(x+y\le80\)
- D) \(x+y\ge80\)
A delivery service charges a fixed fee of $9 plus $4 for each package. A customer has no more than $41 available. If \(p\) is the number of packages, what is the greatest possible whole-number value of \(p\)?
- A) 7
- B) 8
- C) 9
- D) 10
The boundaries of a system are \(y=2x+1\) and \(y=-x+7\). At what point do the two boundary lines intersect?
- A) \((1,3)\)
- B) \((2,5)\)
- C) \((3,7)\)
- D) \((4,9)\)
For \(y\le -2x+4\), which point is in the solution region?
- A) \((0,5)\)
- B) \((1,3)\)
- C) \((2,1)\)
- D) \((3,-3)\)
A student must satisfy \(x+y\le12\) and \(x\ge5\). Which point satisfies both conditions?
- A) \((4,6)\)
- B) \((5,8)\)
- C) \((6,6)\)
- D) \((7,7)\)
Which inequality is equivalent to \(-2x\ge14\)?
- A) \(x\ge-7\)
- B) \(x\le-7\)
- C) \(x\ge7\)
- D) \(x\le7\)
A fundraiser earns $6 for each item sold and has already collected $48. The goal is to collect at least $150. If \(n\) is the number of additional items sold, what is the minimum whole-number value of \(n\)?
- A) 16
- B) 17
- C) 18
- D) 19
Mastery check
Can you solve an inequality without forgetting the negative-sign reversal?
Can you choose a solid or dashed boundary and shade the correct region?
Can you turn an “at least” or “no more than” condition into correct mathematics?
Can you identify the overlap of two inequality regions and check a point against every condition?