Wize High School Grade 12 Pre-Calculus Textbook > Inequalities

Solving Linear Inequalities

Popular Courses

0:00 / 0:00

Solving Linear Inequalities Algebraically
When working with linear inequalities, they can usually be solved with addition, subtraction, multiplication, and/or division in the same way that linear equations are solved. 

One of the biggest differences when working with linear inequalities over linear equations is multiplication and division will affect the direction of the inequality sign. 

Example 1
Let's look at how to solve forxxx for the inequality−x≥1-x\geq1−x≥1.

−x−1≥1−1Divide by -1.x≤−1Change the direction of the inequality sign.\begin{array}{ccl}
\displaystyle-\frac{x}{-1}\geq{}\displaystyle\frac{1}{-1}&&\text{Divide by -1.}\\\\x\leq-1&&\text{Change the direction of the inequality sign.}
\end{array}−−1x​≥−11​x≤−1​​Divide by -1.Change the direction of the inequality sign.​

Wize Tip
When solving inequalities, if multiplying or dividing by a negative number, the direction of the inequality must change. 

PAGE BREAK
Example 2

Solve 2x+3>−112x+3>-112x+3>−11. Represent the solution on a number line. 

2x+3>−11Isolate for x. Subtract 3 on both sides.2x>−14Divide by 2.∴ x>−7\begin{array}{rclcl}
2x+3&>&-11&&\text{Isolate for }x.~\text{Subtract 3 on both sides}.\\\\
2x&>&-14&&\text{Divide by 2.}\\\\
\therefore~x&>&-7
\end{array}2x+32x∴ x​>>>​−11−14−7​​Isolate for x. Subtract 3 on both sides.Divide by 2.​

0:00 / 0:00

Example: Solving Linear Inequalities
Solve 4−3x≤20 4-3x\leq 20~4−3x≤20  for xxx algebraically.

4−3x≤20Subtract 4 from both sides.−3x≤16Divide by -3. Change direction of inequality.∴ x≥−163\begin{array}{rclcl}
4-3x&\leq&20&&\text{Subtract 4 from both sides.}\\\\
-3x&\leq&16&&\text{Divide by -3. Change direction of inequality.}\\\\
\therefore~x&\geq&-\displaystyle\frac{16}{3}
\end{array}4−3x−3x∴ x​≤≤≥​2016−316​​Subtract 4 from both sides.Divide by -3. Change direction of inequality.

0:00 / 0:00

Example: Solving Linear Inequalities 
Solve −2≤6x−1≤10-2\leq{}6x-1\leq{}10−2≤6x−1≤10 for xxx algebraically.

−2≤6x−1≤10Add 1 to both sides of the inequality.−1≤6x≤11Divide both sides by 6 to isolate for x.−16≤x≤116Final Answer. \begin{array}{rccclcl}
-2&\leq&6x-1&\leq&10&&\text{Add 1 to both sides of the inequality.}\\\\
-1&\leq&6x&\leq&11&&\text{Divide both sides by 6 to isolate for }x.\\\\
-\displaystyle\frac{1}{6}&\leq&x&\leq&\displaystyle\frac{11}{6}&&\text{Final Answer. }
\end{array}−2−1−61​​≤≤≤​6x−16xx​≤≤≤​1011611​​​Add 1 to both sides of the inequality.Divide both sides by 6 to isolate for x.Final Answer. ​

Practice: Solving Linear Inequalities

For each inequality, determine if x=−1x=-1x=−1 is a solution. 

A) −x+3≥−2x−5-x+3\geq-2x-5−x+3≥−2x−5 

B) 23−34x<3712+x\displaystyle\frac{2}{3}-\displaystyle\frac{3}{4}x<\displaystyle\frac{37}{12}+x32​−43​x<1237​+x 

C) 8x−2≥5x+168x-2\geq5x+168x−2≥5x+16  

D) (7x−9)(2x+5)>50(7x-9)(2x+5)>50(7x−9)(2x+5)>50  

E) 89x+5<5+4x\displaystyle\frac{8}{9}x+5<5+4x98​x+5<5+4x  

F) 3x−14≤36+2x3x-14\leq36+2x3x−14≤36+2x 

I don't know

Mark Yourself Question

Grab a piece of paper and try this problem yourself.
When you're done, check the "I have answered this question" box below.
View the solution and report whether you got it right or wrong.

Practice: Solving Linear Inequalities

Solve −8≤−3(x−2)≤13-8\leq{}-3(x-2)\leq{}13−8≤−3(x−2)≤13 for x.x.x.

I have answered this question

Practice: Solving Linear Inequalities
Solve ∣x−1∣≥2|x-1|\geq{}2∣x−1∣≥2. Express the answer in interval notation. 

A) (−∞,−1) ∪ (3,∞)(-\infin,-1)~\cup~(3,\infin)(−∞,−1) ∪ (3,∞) 

B) (−∞,−1] ∪ [3,∞)(-\infin,-1]~\cup~[3,\infin)(−∞,−1] ∪ [3,∞) 

C) (−∞,−3) ∪ (1,∞)(-\infin,-3)~\cup~(1,\infin)(−∞,−3) ∪ (1,∞) 

D) (−∞,−3] ∪ [1,∞)(-\infin,-3]~\cup~[1,\infin)(−∞,−3] ∪ [1,∞) 

I don't know

Extra Practice

Solving Linear Inequalities

Practice: Solving Linear Inequalities
Solve ∣2x+3∣<10x−7|2x+3|<10x-7∣2x+3∣<10x−7. Express the answer in interval notation. 

Solving Linear Inequalities

Practice: Solving Linear Inequalities
Solve ∣x+3∣≥5|x+3|\geq{}5∣x+3∣≥5. Express the answer in interval notation. 

Solving Linear Inequalities

Practice: Solving Linear Inequalities
Solve 1≤−35(25x+10)+9≤151\leq{}-\dfrac{3}{5}(25x+10)+9\leq{}151≤−53​(25x+10)+9≤15 for x.x.x.

Solving Linear Inequalities

Practice: Solving Linear Inequalities
Solve −2≤−4(x+3)≤7-2\leq{}-4(x+3)\leq{}7−2≤−4(x+3)≤7 for x.x.x.

Solving Linear Inequalities

Practice: Solving Linear Inequalities
For each inequality, determine if x=2x=2x=2 is a solution. 

Solving Linear Inequalities

Practice: Solving Linear Inequalities
For each inequality, determine if x=1x=1x=1 is a solution. 