Solving multi-step inequalities involves using inverse operations to isolate the variable, just like solving equations. You may need to distribute, combine like terms, or move variables from one side to the other. The most important rule is that multiplying or dividing both sides by a negative number reverses the direction of the inequality sign. Solutions can be expressed using inequality notation and represented on a number line.
Notes
Problems
Solve. Express as an inequality.
\(\textbf{1)}\) \(-5x\le20\) The answer is \(x\ge-4\), see video for graph
\(\,\,\,\,\,\,-5x\le20\)
Divide both sides by -5 and reverse the inequality sign.
\(\,\,\,\,\,\,\frac{-5x}{-5}\ge\frac{20}{-5}\)
\(\,\,\,\,\,\,x\ge-4\)
\(\,\,\,\,\,\,-5x\le20\)
Divide both sides by -5 and reverse the inequality sign.
\(\,\,\,\,\,\,\frac{-5x}{-5}\ge\frac{20}{-5}\)
\(\,\,\,\,\,\,x\ge-4\)
\(\textbf{2)}\) \(5x-4\gt2x-10\) The answer is \(x\gt-2\), see video for graph
\(\,\,\,\,\,\,5x-4\gt2x-10\)
Subtract 2x from both sides.
\(\,\,\,\,\,\,3x-4\gt-10\)
Add 4 to both sides.
\(\,\,\,\,\,\,3x\gt-6\)
Divide both sides by 3.
\(\,\,\,\,\,\,x\gt-2\)
\(\,\,\,\,\,\,5x-4\gt2x-10\)
Subtract 2x from both sides.
\(\,\,\,\,\,\,3x-4\gt-10\)
Add 4 to both sides.
\(\,\,\,\,\,\,3x\gt-6\)
Divide both sides by 3.
\(\,\,\,\,\,\,x\gt-2\)
\(\textbf{3)}\) \(18-4x\lt2\) The answer is \(x\gt4\), see video for graph
\(\,\,\,\,\,\,18-4x\lt2\)
Subtract 18 from both sides.
\(\,\,\,\,\,\,-4x\lt-16\)
Divide both sides by -4 and reverse the inequality sign.
\(\,\,\,\,\,\,x\gt4\)
\(\,\,\,\,\,\,18-4x\lt2\)
Subtract 18 from both sides.
\(\,\,\,\,\,\,-4x\lt-16\)
Divide both sides by -4 and reverse the inequality sign.
\(\,\,\,\,\,\,x\gt4\)
\(\textbf{4)}\) \(5-2x\lt11\) The answer is \(x\gt-3\), see video for graph
\(\,\,\,\,\,\,5-2x\lt11\)
Subtract 5 from both sides.
\(\,\,\,\,\,\,-2x\lt6\)
Divide both sides by -2 and reverse the inequality sign.
\(\,\,\,\,\,\,x\gt-3\)
\(\,\,\,\,\,\,5-2x\lt11\)
Subtract 5 from both sides.
\(\,\,\,\,\,\,-2x\lt6\)
Divide both sides by -2 and reverse the inequality sign.
\(\,\,\,\,\,\,x\gt-3\)
\(\textbf{5)}\) \(2-5x\ge6(x-5)+1\) The answer is \(x\le\frac{31}{11}\), see video for graph
\(\,\,\,\,\,\,2-5x\ge6(x-5)+1\)
Distribute the 6.
\(\,\,\,\,\,\,2-5x\ge6x-30+1\)
Combine like terms.
\(\,\,\,\,\,\,2-5x\ge6x-29\)
Subtract 6x from both sides.
\(\,\,\,\,\,\,2-11x\ge-29\)
Subtract 2 from both sides.
\(\,\,\,\,\,\,-11x\ge-31\)
Divide both sides by -11 and reverse the inequality sign.
\(\,\,\,\,\,\,x\le\frac{31}{11}\)
\(\,\,\,\,\,\,2-5x\ge6(x-5)+1\)
Distribute the 6.
\(\,\,\,\,\,\,2-5x\ge6x-30+1\)
Combine like terms.
\(\,\,\,\,\,\,2-5x\ge6x-29\)
Subtract 6x from both sides.
\(\,\,\,\,\,\,2-11x\ge-29\)
Subtract 2 from both sides.
\(\,\,\,\,\,\,-11x\ge-31\)
Divide both sides by -11 and reverse the inequality sign.
\(\,\,\,\,\,\,x\le\frac{31}{11}\)
\(\textbf{6)}\) \(\frac{x}{5}+2\le-10\) The answer is \(x\le-60\), see video for graph
\(\,\,\,\,\,\,\frac{x}{5}+2\le-10\)
Subtract 2 from both sides.
\(\,\,\,\,\,\,\frac{x}{5}\le-12\)
Multiply both sides by 5.
\(\,\,\,\,\,\,x\le-60\)
\(\,\,\,\,\,\,\frac{x}{5}+2\le-10\)
Subtract 2 from both sides.
\(\,\,\,\,\,\,\frac{x}{5}\le-12\)
Multiply both sides by 5.
\(\,\,\,\,\,\,x\le-60\)
\(\textbf{7)}\) \(5x-4\le2x+2\) The answer is \(x\le2\)
\(\,\,\,\,\,\,5x-4\le2x+2\)
Subtract 2x from both sides.
\(\,\,\,\,\,\,3x-4\le2\)
Add 4 to both sides.
\(\,\,\,\,\,\,3x\le6\)
Divide both sides by 3.
\(\,\,\,\,\,\,x\le2\)
\(\,\,\,\,\,\,5x-4\le2x+2\)
Subtract 2x from both sides.
\(\,\,\,\,\,\,3x-4\le2\)
Add 4 to both sides.
\(\,\,\,\,\,\,3x\le6\)
Divide both sides by 3.
\(\,\,\,\,\,\,x\le2\)
\(\textbf{8)}\) \(4-x\le8\) The answer is \(x\ge-4\)
\(\,\,\,\,\,\,4-x\le8\)
Subtract 4 from both sides.
\(\,\,\,\,\,\,-x\le4\)
Divide both sides by -1 and reverse the inequality sign.
\(\,\,\,\,\,\,x\ge-4\)
\(\,\,\,\,\,\,4-x\le8\)
Subtract 4 from both sides.
\(\,\,\,\,\,\,-x\le4\)
Divide both sides by -1 and reverse the inequality sign.
\(\,\,\,\,\,\,x\ge-4\)
\(\textbf{9)}\) \(5x+1\gt11\) The answer is \(x\gt2\)
\(\,\,\,\,\,\,5x+1\gt11\)
Subtract 1 from both sides.
\(\,\,\,\,\,\,5x\gt10\)
Divide both sides by 5.
\(\,\,\,\,\,\,x\gt2\)
\(\,\,\,\,\,\,5x+1\gt11\)
Subtract 1 from both sides.
\(\,\,\,\,\,\,5x\gt10\)
Divide both sides by 5.
\(\,\,\,\,\,\,x\gt2\)
\(\textbf{10)}\) \(x\le2x-2\) The answer is \(x\ge2\)
\(\,\,\,\,\,\,x\le2x-2\)
Subtract 2x from both sides.
\(\,\,\,\,\,\,-x\le-2\)
Divide both sides by -1 and reverse the inequality sign.
\(\,\,\,\,\,\,x\ge2\)
\(\,\,\,\,\,\,x\le2x-2\)
Subtract 2x from both sides.
\(\,\,\,\,\,\,-x\le-2\)
Divide both sides by -1 and reverse the inequality sign.
\(\,\,\,\,\,\,x\ge2\)
\(\textbf{11)}\) Express as an inequality. The sum of a number and 4 is at least 10. The answer is \(x+4\ge10\), see video for graph
Let x represent the number.
“The sum of a number and 4” means add 4 to x.
\(\,\,\,\,\,\,x+4\)
“At least 10” means greater than or equal to 10.
\(\,\,\,\,\,\,x+4\ge10\)
Let x represent the number.
“The sum of a number and 4” means add 4 to x.
\(\,\,\,\,\,\,x+4\)
“At least 10” means greater than or equal to 10.
\(\,\,\,\,\,\,x+4\ge10\)
\(\textbf{12)}\) Express as an inequality. The product of a number and 4 is less than 10. The answer is \(4x\lt10\), see video for graph
Let x represent the number.
“The product of a number and 4” means multiply x by 4.
\(\,\,\,\,\,\,4x\)
“Less than 10” means use the less-than symbol.
\(\,\,\,\,\,\,4x\lt10\)
Let x represent the number.
“The product of a number and 4” means multiply x by 4.
\(\,\,\,\,\,\,4x\)
“Less than 10” means use the less-than symbol.
\(\,\,\,\,\,\,4x\lt10\)
\(\textbf{13)}\) Express as an inequality. Half of x is at most 7. The answer is \(\frac{x}{2}\le7\)
“Half of x” means divide x by 2.
\(\,\,\,\,\,\,\frac{x}{2}\)
“At most 7” means less than or equal to 7.
\(\,\,\,\,\,\,\frac{x}{2}\le7\)
“Half of x” means divide x by 2.
\(\,\,\,\,\,\,\frac{x}{2}\)
“At most 7” means less than or equal to 7.
\(\,\,\,\,\,\,\frac{x}{2}\le7\)
\(\textbf{14)}\) Express as an inequality. The sum of x and 3 is greater than the sum of 3x and 2. The answer is \(x+3\gt3x+2\)
“The sum of x and 3” means add x and 3.
\(\,\,\,\,\,\,x+3\)
“The sum of 3x and 2” means add 3x and 2.
\(\,\,\,\,\,\,3x+2\)
“Is greater than” means use the greater-than symbol.
\(\,\,\,\,\,\,x+3\gt3x+2\)
“The sum of x and 3” means add x and 3.
\(\,\,\,\,\,\,x+3\)
“The sum of 3x and 2” means add 3x and 2.
\(\,\,\,\,\,\,3x+2\)
“Is greater than” means use the greater-than symbol.
\(\,\,\,\,\,\,x+3\gt3x+2\)
\(\textbf{15)}\) Express as an inequality. The product of x and 3 is less than the sum of x and 3. The answer is \(3x\lt x+3\)
“The product of x and 3” means multiply x by 3.
\(\,\,\,\,\,\,3x\)
“The sum of x and 3” means add x and 3.
\(\,\,\,\,\,\,x+3\)
“Is less than” means use the less-than symbol.
\(\,\,\,\,\,\,3x\lt x+3\)
“The product of x and 3” means multiply x by 3.
\(\,\,\,\,\,\,3x\)
“The sum of x and 3” means add x and 3.
\(\,\,\,\,\,\,x+3\)
“Is less than” means use the less-than symbol.
\(\,\,\,\,\,\,3x\lt x+3\)
More Practice Problems
\(\textbf{16)}\) \(3(x-4)+5\le2x+1\) The answer is \(x\le8\)
\(\,\,\,\,\,\,3(x-4)+5\le2x+1\)
Distribute the 3.
\(\,\,\,\,\,\,3x-12+5\le2x+1\)
Combine like terms.
\(\,\,\,\,\,\,3x-7\le2x+1\)
Subtract 2x from both sides.
\(\,\,\,\,\,\,x-7\le1\)
Add 7 to both sides.
\(\,\,\,\,\,\,x\le8\)
\(\,\,\,\,\,\,3(x-4)+5\le2x+1\)
Distribute the 3.
\(\,\,\,\,\,\,3x-12+5\le2x+1\)
Combine like terms.
\(\,\,\,\,\,\,3x-7\le2x+1\)
Subtract 2x from both sides.
\(\,\,\,\,\,\,x-7\le1\)
Add 7 to both sides.
\(\,\,\,\,\,\,x\le8\)
\(\textbf{17)}\) \(4-2(3x-1)\gt18\) The answer is \(x\lt-2\)
\(\,\,\,\,\,\,4-2(3x-1)\gt18\)
Distribute the -2.
\(\,\,\,\,\,\,4-6x+2\gt18\)
Combine like terms.
\(\,\,\,\,\,\,6-6x\gt18\)
Subtract 6 from both sides.
\(\,\,\,\,\,\,-6x\gt12\)
Divide both sides by -6 and reverse the inequality sign.
\(\,\,\,\,\,\,x\lt-2\)
\(\,\,\,\,\,\,4-2(3x-1)\gt18\)
Distribute the -2.
\(\,\,\,\,\,\,4-6x+2\gt18\)
Combine like terms.
\(\,\,\,\,\,\,6-6x\gt18\)
Subtract 6 from both sides.
\(\,\,\,\,\,\,-6x\gt12\)
Divide both sides by -6 and reverse the inequality sign.
\(\,\,\,\,\,\,x\lt-2\)
\(\textbf{18)}\) \(\frac{x-3}{4}+2\ge5\) The answer is \(x\ge15\)
\(\,\,\,\,\,\,\frac{x-3}{4}+2\ge5\)
Subtract 2 from both sides.
\(\,\,\,\,\,\,\frac{x-3}{4}\ge3\)
Multiply both sides by 4.
\(\,\,\,\,\,\,x-3\ge12\)
Add 3 to both sides.
\(\,\,\,\,\,\,x\ge15\)
\(\,\,\,\,\,\,\frac{x-3}{4}+2\ge5\)
Subtract 2 from both sides.
\(\,\,\,\,\,\,\frac{x-3}{4}\ge3\)
Multiply both sides by 4.
\(\,\,\,\,\,\,x-3\ge12\)
Add 3 to both sides.
\(\,\,\,\,\,\,x\ge15\)
\(\textbf{19)}\) \(0.5x+3\lt0.2x+12\) The answer is \(x\lt30\)
\(\,\,\,\,\,\,0.5x+3\lt0.2x+12\)
Subtract 0.2x from both sides.
\(\,\,\,\,\,\,0.3x+3\lt12\)
Subtract 3 from both sides.
\(\,\,\,\,\,\,0.3x\lt9\)
Divide both sides by 0.3.
\(\,\,\,\,\,\,x\lt30\)
\(\,\,\,\,\,\,0.5x+3\lt0.2x+12\)
Subtract 0.2x from both sides.
\(\,\,\,\,\,\,0.3x+3\lt12\)
Subtract 3 from both sides.
\(\,\,\,\,\,\,0.3x\lt9\)
Divide both sides by 0.3.
\(\,\,\,\,\,\,x\lt30\)
\(\textbf{20)}\) \(2(x+3)\le2x+1\) The answer is no solution.
\(\,\,\,\,\,\,2(x+3)\le2x+1\)
Distribute the 2.
\(\,\,\,\,\,\,2x+6\le2x+1\)
Subtract 2x from both sides.
\(\,\,\,\,\,\,6\le1\)
This statement is false.
Therefore, there is no solution.
\(\,\,\,\,\,\,2(x+3)\le2x+1\)
Distribute the 2.
\(\,\,\,\,\,\,2x+6\le2x+1\)
Subtract 2x from both sides.
\(\,\,\,\,\,\,6\le1\)
This statement is false.
Therefore, there is no solution.
\(\textbf{21)}\) \(3(x-2)\gt3x-10\) The answer is all real numbers.
\(\,\,\,\,\,\,3(x-2)\gt3x-10\)
Distribute the 3.
\(\,\,\,\,\,\,3x-6\gt3x-10\)
Subtract 3x from both sides.
\(\,\,\,\,\,\,-6\gt-10\)
This statement is always true.
Therefore, all real numbers are solutions.
\(\,\,\,\,\,\,3(x-2)\gt3x-10\)
Distribute the 3.
\(\,\,\,\,\,\,3x-6\gt3x-10\)
Subtract 3x from both sides.
\(\,\,\,\,\,\,-6\gt-10\)
This statement is always true.
Therefore, all real numbers are solutions.
See Related Pages\(\)
\(\bullet\text{ Inequality Calculator}\)
\(\,\,\,\,\,\,\,\,\text{(Symbolab.com)}\)
\(\bullet\text{ Solving Inequalities Addition and Subtraction}\)
\(\,\,\,\,\,\,\,\,4+x\ge12…\)
\(\bullet\text{ Solving Inequalities Multiplication and Division}\)
\(\,\,\,\,\,\,\,\,5x\lt-45…\)
\(\bullet\text{ Solving Compound Inequalities}\)
\(\,\,\,\,\,\,\,\,3\lt6-2x\le18…\)