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

Solving Inequalities Involving Absolute Values

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Solving Inequalities with Absolute Values



Absolute Value

When we have a variable inside of an absolute value, and set equal to a given number, there are two possible solutions.

∣x∣=3x=−3    x=3|x| = 3 \\ x = -3 \text{ \ \ \ } x = 3
This comes from the piecewise definition of an inequality.

∣x∣={−x,x<0x,x≥0|x|=\begin{cases} -x&,&x<0\\ x&,&x\geq0 \end{cases}

Example 1
Solve: 12∣5x+2∣=6\frac{1}{2}|5x + 2| = 6

ANSWER:
First we work to isolate the absolute value on one side

12∣5x+2∣=6∣5x+2∣=12\begin{aligned} \frac{1}{2}|5x + 2| &= 6 \\ |5x + 2| &= 12 \end{aligned}
Now we can split into the two possibilities.

∣5x+2∣=12−(5x+2)=12  and  5x+2=12\begin{aligned} |5x + 2| &= 12 \\ -(5x + 2)&= 12 \text{ \ and \ } 5x + 2 &= 12 \end{aligned}
We can now solve each of these equations separately.

−(5x+2)=12  and  5x+2=125x+2=−12  and  5x=105x=−14  and  x=2x=−145  \begin{array}{rcl} -(5x + 2) = 12 &\text{ \ and \ } & 5x + 2 = 12 \\ 5x + 2 = -12 &\text{ \ and \ }& 5x = 10 \\ 5x = -14 &\text{ \ and \ }& x = 2 \\ x = \displaystyle\frac{-14}{5} &\text{ \ } & \end{array}


Combining with Inequalities

When solving an inequality with an absolute value, we still must consider two cases. In addition we must remember to flip the sign and how to combine the inequalities.
  • ∣ expression ∣<number| \colorThree{\text{ expression }} | < \colorFive{ \text{number}} connect using AND
  • ∣ expression ∣>number| \colorThree{\text{ expression }} | > \colorFive{ \text{number}} connect using OR
Example 2
Solve: ∣3x+2∣<5|3x + 2| < 5

Since the absolute value is already isolated, we'll split into two cases connected using AND

∣3x+2∣<5−(3x+2)<5 AND 3x+2<5\begin{aligned} |3x + 2| &< 5 \\ -(3x + 2) &< 5 \text{ AND } 3x + 2 < 5 \end{aligned}
We now solve each of these separately

−(3x+2)<5 AND 3x+2<53x+2>−5 AND 3x<33x>−7 AND x<1x>−73 AND x<1\begin{array}{rcl} -(3x + 2) < 5 &\text{ AND }& 3x + 2 < 5 \\ 3x + 2 > -5 &\text{ AND }& 3x < 3 \\ 3x > -7 &\text{ AND }& x < 1 \\ x > \displaystyle\frac{-7}{3} &\text{ AND }& x < 1 \\ \end{array}
We can also write this as an interval (−73,1)\displaystyle\left( \frac{-7}{3}, 1\right)
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Temperature of Venus



The average surface temperature on Venus varies from day to night. The range of temperatures in degrees Fahrenheit can be approximated by the following inequality

3∣x−860∣≤1203|x - 860 | \leq 120

What is the range of temperatures found on Venus? Express your answer as an interval.

ANSWER:
First we work to isolate the absolute value

3∣x−860∣≤120∣x−860∣≤40\begin{aligned} 3|x - 860 | &\leq 120 \\ |x - 860| &\leq 40 \end{aligned}
Now we split into two cases

−(x−860)≤40ANDx−860≤40x−860≥−40ANDx≤900x≥820ANDx≤900\begin{array}{lcr} -(x-860) \leq 40 & AND & x - 860 \leq 40 \\ x - 860 \geq - 40 & AND & x \leq 900 \\ x \geq 820 & AND & x \leq 900 \end{array}
So the temperature varies between 820 and 900 degrees Fahrenheit.
As an interval [820,900][820, 900]

Practice: Solving Linear Inequalities

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