Absolute Value Inequalities
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Question 1 of 4
1. Question
Solve for `x``|2x+7|≤5`Hint
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Representing Inequalities in the Number Line
Greater than (`>`)
Greater than or equal (`≥`)
Less than (`<`)
Less than or equal (`≤`)
First, since we are solving an absolute value equation, form a positive and negative equation and solve for `x` on both equations.For the negative value, switch the inequality sign.Positive:`2x+7` `≤` `5` `2x+7` `-7` `≤` `5` `-7` Subtract `7` from both sides `2x` `≤` `-2` `2x` `div2` `≤` `-2` `div2` Divide both sides by `2` `x` `≤` `-1` Negative:`2x+7` `≥` `-5` `2x+7` `-7` `≥` `-5` `-7` Subtract `7` from both sides `2x` `≥` `-12` `2x` `div2` `≥` `-12` `div2` Divide both sides by `2` `x` `≥` `-6` Next, plot the values of `x` on the number line and draw the inequality.
Since the inequality has a less than or equal to sign, `x` should be between the known values.Finally, write the value of `x` with the variable in the middle and the known values on each sideKnown values`x` `≤` `-1` `-6` `≤` `x` Same as `x≥-6` `-6` `≤` `x` `≤` `-1` `-6≤x≤-1` -
Question 2 of 4
2. Question
Plot the inequality`|(2-3x)/4|>7`Hint
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Representing Inequalities in the Number Line
Greater than (`>`)Greater than or equal (`≥`)Less than (`<`)Less than or equal (`≤`)First, since we are solving an absolute value equation, form a positive and negative equation and solve for `x` on both equations.For the negative value, switch the inequality sign.Positive:`(2-3x)/4` `>` `7` `(2-3x)/4` `times4` `>` `7` `times4` Multiply both sides by `4` `2-3x` `>` `28` `2-3x` `-2` `>` `28` `-2` Subtract `2` from both sides `-3x` `>` `26` `-3x` `div(-3)` `>` `26` `div(-3)` Divide both sides by `-3` `x` `<` `-(26)/3` Dividing both sides by a negative value reverses the inequality `x` `<` `-8 2/3` Convert to a mixed number Negative:`(2-3x)/4` `<` `-7` `(2-3x)/4` `times4` `<` `-7` `times4` Multiply both sides by `4` `2-3x` `<` `-28` `2-3x` `-2` `<` `-28` `-2` Subtract `2` from both sides `-3x` `<` `-30` `-3x` `div(-3)` `>` `-30` `div(-3)` Divide both sides by `-3` `x` `>` `10` Dividing both sides by a negative value reverses the inequality Next, plot the values of `x` on the number line and draw the inequality.
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Question 3 of 4
3. Question
Solve for `x``6|1/2 x+5|<6`Hint
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Keep Going!
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Representing Inequalities in the Number Line
Greater than (`>`)Greater than or equal (`≥`)Less than (`<`)Less than or equal (`≤`)First, convert the equation to its standard form`6|1/2 x+5|` `<` `6` `6|1/2 x+5|` `div6` `<` `6` `div6` Divide both sides by `6` `|1/2 x+5|` `<` `1` Next, since we are solving an absolute value equation, form a positive and negative equation and solve for `x` on both equations.For the negative value, switch the inequality sign.Positive:`1/2 x+5` `<` `1` `1/2 x+5` `-5` `<` `1` `-5` Subtract `5` from both sides `1/2 x` `<` `-4` `1/2 x` `times2` `<` `-4` `times2` Multiply both sides by `2` `x` `<` `-8` Negative:`1/2 x+5` `>` `-1` `1/2 x+5` `-5` `>` `-1` `-5` Subtract `5` from both sides `1/2 x` `>` `-6` `1/2 x` `times2` `>` `-6` `times2` Multiply both sides by `2` `x` `>` `-12` Next, plot the values of `x` on the number line and draw the inequality.
Since the inequality has a less than sign, `x` should be between the known values.Finally, write the value of `x` with the variable in the middle and the known values on each sideKnown values`x` `<` `-8` `-12` `<` `x` Same as `x``>``-12` `-12` `<` `x` `<` `-8` `-12``<``x``<``-8` -
Question 4 of 4
4. Question
Plot the inequality`2|1/3 y-4|-3>7`Hint
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Correct!
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Representing Inequalities in the Number Line
Greater than (`>`)Greater than or equal (`≥`)Less than (`<`)Less than or equal (`≤`)First, convert the equation to its standard form`2|1/3 y-4|-3` `>` `7` `2|1/3 y-4|-3` `+3` `>` `7` `+3` Add `3` to both sides `2|1/3 y-4|` `>` `10` `2|1/3 y-4|` `div2` `>` `10` `div2` Divide both sides by `2` `|1/3 y-4|` `>` `5` Next, since we are solving an absolute value equation, form a positive and negative equation and solve for `x` on both equations.For the negative value, switch the inequality sign.Positive:`1/3 y-4` `>` `5` `1/3 y-4` `+4` `>` `5` `+4` Add `4` to both sides `1/3 y` `>` `9` `1/3 y` `times3` `>` `9` `times3` Multiply both sides by `3` `y` `>` `27` Negative:`1/3 y-4` `<` `-5` `1/3 y-4` `+4` `<` `-5` `+4` Add `4` to both sides `1/3 y` `<` `-1` `1/3 y` `times3` `<` `-1` `times3` Multiply both sides by `3` `y` `<` `-3` Next, plot the values of `x` on the number line and draw the inequality.