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Wize High School Grade 9 Math Textbook > Solving 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.
  1. Simplify the left side and right side of the inequality sign separately
  2. Expand and simplify any brackets
  3. Group like terms
  4. Deal with   ...   +3\boxed{~~...~~}~\bcth{+3} and   ...   −3\boxed{~~...~~}~\bcth{-3} first
  5. The opposite of addition is subtraction
  6. The opposite of subtraction is addition
  7. Whatever you do to one side, you MUST do to the other
  8. Deal with ×3   ...  \bcfi{\times3}~\boxed{~~...~~} and   ...  3\bcfi{\dfrac{\boxed{~~...~~}}{3}} next
  9. The opposite of multiplication is division
  10. The opposite of division is multiplication
  11. Whatever you do to one side, you MUST do to the other
  12. When you multiply or divide by a negative number, you must FLIP THE INEQUALITY SIGN
  13. >> becomes <<
  14. << becomes >>
  15. ≥\ge becomes ≤\le
  16. ≤\le becomes ≥\ge

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Example: Solving Inequalities

Solve the following inequalities.

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

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

b) 3x≥−153x\ge-15

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

c) x2>8\dfrac{x}{2}>8

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

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d) −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}
*Make sure to flip the inequality sign when you multiple or divide by a negative number.!


e) −2x3≥−10-\dfrac{2x}{3}\ge-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}

Practice: Solving Inequalities

Solve the following inequalities.
x+5.5≤13.4x+5.5\le13.4

Practice: Solving Inequalities

Solve the following inequalities.
6x≥−156x\ge-15
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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).

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}

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}

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}
The solution to this inequality is y≥1514\boxed{y\ge\dfrac{15}{14}}.

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} we could have used a division to solve for yy:
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}

Practice: Solving Inequalities

Solve the inequality 2x<5x−182x<5x-18.

Practice: Solving Inequalities

Solve the inequality 14(x−2)≥x−7\dfrac{1}{4}(x-2)\ge x-7.
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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?