Wize High School Algebra I Textbook (Common Core) > Solving One Variable Inequalities

Solving Inequalities

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Solving Inequalities
To solve an inequality, we follow the same steps as solving an equation, but with a few changes.
Simplify the left side and right side of the inequality sign separately
Expand and simplify any brackets
Group like terms
Deal with   ...   +3\boxed{~~...~~}~\bcth{+3}  ...  ​ +3 and   ...   −3\boxed{~~...~~}~\bcth{-3}  ...  ​ −3 first
The opposite of addition is subtraction
The opposite of subtraction is addition
Whatever you do to one side, you MUST do to the other
Deal with ×3   ...  \bcfi{\times3}~\boxed{~~...~~}×3   ...  ​ and   ...  3\bcfi{\dfrac{\boxed{~~...~~}}{3}}3  ...  ​​ next
The opposite of multiplication is division
The opposite of division is multiplication
Whatever you do to one side, you MUST do to the other
When you multiply or divide by a negative number, you must FLIP THE INEQUALITY SIGN
>>> becomes <<<
<<< becomes >>>
≥\ge≥ becomes ≤\le≤
≤\le≤ becomes ≥\ge≥

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Example: Solving Inequalities
Solve the following inequalities.

a) x+5>6x+5>6x+5>6

x+5−5>6−5x>1\begin{array}{rcl}
x+5\colorTwo{-5} &>& 6\colorTwo{-5}\\\\
x&>&1
\end{array}x+5−5x​>>​6−51​

b) 3x≥−153x\ge-153x≥−15

3x3≥−153x≥−5\begin{array}{rcl}
\dfrac{3x}{\colorTwo{3}} &\ge& \dfrac{-15}{\colorTwo{3}}\\\\
x&\ge&-5
\end{array}33x​x​≥≥​3−15​−5​

c) x2>8\dfrac{x}{2}>82x​>8

x2×2>8×2x>16\begin{array}{rcl}
\dfrac{x}{2}\colorTwo{\times2} &>& 8\colorTwo{\times2}\\
x&>&16
\end{array}2x​×2x​>>​8×216​

PAGE BREAK
d) −2x<10-2x<10−2x<10

−2x−2<10−2x>−5\begin{array}{rcl}
\dfrac{-2x}{\colorTwo{-2}} &<& \dfrac{10}{\colorTwo{-2}}\\\\
x&\colorbox{yellow}>&-5
\end{array}−2−2x​x​<>​​−210​−5​
*Make sure to flip the inequality sign when you multiple or divide by a negative number.!

e) −2x3≥−10-\dfrac{2x}{3}\ge-10−32x​≥−10

−2x3−23≥−10−23x≤−10×−32x≤15\begin{array}{rcl}
\dfrac{-\dfrac{2x}{3}}{\scriptsize\colorTwo{-\dfrac{2}{3}}} &\ge& \dfrac{-10}{\scriptsize\colorTwo{-\dfrac{2}{3}}}\\\\
x&\colorbox{yellow}{$\le$}&-10\times\scriptsize\colorTwo{-\dfrac{3}{2}}\\\\
x&\le&15
\end{array}−32​−32x​​xx​≥≤​≤​−32​−10​−10×−23​15​

Practice: Solving Inequalities
Solve the following inequalities.

Practice: Solving Inequalities
Solve the following inequalities.

6x≥−156x\ge-156x≥−15

A) x≥−52x\ge-\dfrac{5}{2}x≥−25​

B) x≤−52x\le-\dfrac{5}{2}x≤−25​

C) x>−52x>-\dfrac{5}{2}x>−25​

D) x<−52x<-\dfrac{5}{2}x<−25​

E) x>52x>\dfrac{5}{2}
x>25​

I don't know

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Example: Solving Inequalities
Solve the inequality 13y−1≥12(3−4y)\dfrac{1}{3}y-1\ge\dfrac{1}{2}\left(3-4y\right)31​y−1≥21​(3−4y).

Simplify each side of the inequality as much as possible:
13y−1≥12(3)+12(−4y)13y−1≥32−2y\begin{array}{rcl}
\dfrac{1}{3}y-1&\ge&\dfrac{1}{2}(3)+\dfrac{1}{2}(-4y)\\[1em]
\dfrac{1}{3}y-1&\ge&\dfrac{3}{2}-2y\\


\end{array}31​y−131​y−1​≥≥​21​(3)+21​(−4y)23​−2y​

Move all of the variables to one side of the inequality (it doesn't matter which side), and the constants (numbers) to the other side:
13y−1≥32−2y+2y            +2y13y+2y−1≥32+1≥+113y+2y≥32+1\begin{array}{rcl} 
\dfrac{1}{3}y-1&\ge&\dfrac{3}{2}-2y\\
\colorTwo{+2y}~~~~~~~&&~~~~~\colorTwo{+2y}\\[1em]
\dfrac{1}{3}y+2y-1&\ge&\dfrac{3}{2}\\
\colorTwo{+1}&\ge&\colorTwo{+1}\\[1em]
\dfrac{1}{3}y+2y&\ge&\dfrac{3}{2}+1\\[1em]

\end{array}
31​y−1+2y       31​y+2y−1+131​y+2y​≥≥≥≥​23​−2y     +2y23​+123​+1​

Simplify each side of the inequality by finding a common denominator:
13y+63y≥32+2273y≥5273y×3≥52×37y≥1527y7≥1527y≥152×17y≥1514\begin{array}{rcl}
\dfrac{1}{3}y+\dfrac{6}{3}y&\ge&\dfrac{3}{2}+\dfrac{2}{2}\\[1em]
\dfrac{7}{3}y&\ge&\dfrac{5}{2}\\[1em]
\dfrac{7}{3}y\colorTwo{\times3}&\ge&\dfrac{5}{2}\colorTwo{\times3}\\[1em]
7y&\ge&\dfrac{15}{2}\\[1em]
\dfrac{7y}{\colorTwo{7}}&\ge&\dfrac{\dfrac{15}{2}}{\colorTwo{7}}\\[1em]
y&\ge&\dfrac{15}{2}\times\dfrac{1}{7}\\[1em]
y&\ge&\dfrac{15}{14}
\end{array}31​y+36​y37​y37​y×37y77y​yy​≥≥≥≥≥≥≥​23​+22​25​25​×3215​7215​​215​×71​1415​​
The solution to this inequality is y≥1514\boxed{y\ge\dfrac{15}{14}}y≥1415​​.

Alternatively
When we got to this step of the inequality 73y≥52\begin{array}{rcl}

\dfrac{7}{3}y&\ge&\dfrac{5}{2}\\[1em]

\end{array}37​y​≥​25​​ we could have used a division to solve for yyy:
73y≥5273y73≥5273y≥52×37y≥1514\begin{array}{rcl}

\dfrac{7}{3}y&\ge&\dfrac{5}{2}\\[1em]

\dfrac{\dfrac{7}{3}y}{\colorTwo{\frac{7}{3}}}&\ge&\dfrac{\dfrac{5}{2}}{\colorTwo{\frac{7}{3}}}\\[1em]

y&\ge&\dfrac{5}{2}\times\dfrac{3}{7}\\[1em]

y&\ge&\dfrac{15}{14}
\end{array}37​y37​37​y​yy​≥≥≥≥​25​37​25​​25​×73​1415​​

Practice: Solving Inequalities
Solve the inequality 2x<5x−182x<5x-182x<5x−18.

Practice: Solving Inequalities
Solve the inequality 14(x−2)≥x−7\dfrac{1}{4}(x-2)\ge 
x-741​(x−2)≥x−7.

\ge

is entered as >= and

\le

is entered as <=

I don't know

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Practice: Solving Inequalities
The cell phone plan from Bido charges a monthly fee of $5 plus $0.10 per text message sent. The cell phone plan from Breedom mobile doesn't charge a monthly fee, but charges $0.15 per text message sent. How many text messages must be sent  to make the cell phone plan from Bido less expensive than the plan from Breedom mobile?

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