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

Solving Compound Inequalities

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Solving Compound Inequalities

Compound Inequalities

Two inequalities can be combined with each other to form a compound inequality. For these there is a common expression that allows all parts to be compared with each other.

Solving works much the same as a regular linear equality. Try and keep in mind the following:
  • What ever you do to one piece, you must do to all the pieces
  • If you multiply or divide by a negative number, both inequality symbols flip

Example:

Solve −3≤2x+5 And 2x+5<7-3 \leq 2x + 5 \text{ And } 2x + 5 < 7

ANSWER:
We can first combine the inequalities into a compound inequality.

−3≤2x+5 And 2x+5<7−3≤2x+5<7\begin{aligned} -3 &\leq 2x + 5 \text{ And } 2x + 5 < 7 \\ -3 &\leq 2x + 5 < 7 \end{aligned}
From here we work to isolate the xx variable.

−3≤2x+5<7−8≤2x<2−4≤x<1\begin{aligned} -3 &\leq 2x + 5 & < 7 \\ -8 & \leq 2x & < 2 \\ -4 & \leq x & < 1 \end{aligned}
So xx is a number between -4 and 1. Note that the -4 is included in the solution, but 1 is not. We can also express the answer as an interval:
x∈[−4,1)x \in [-4, 1)
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Example: Solving Linear Inequalities

Solve −2≤6x−1≤10-2\leq{}6x-1\leq{}10 for xx 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}

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Practice: Solving Linear Inequalities


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