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`(23x)/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:`(23x)/4` `>` `7` `(23x)/4` `times4` `>` `7` `times4` Multiply both sides by `4` `23x` `>` `28` `23x` `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:`(23x)/4` `<` `7` `(23x)/4` `times4` `<` `7` `times4` Multiply both sides by `4` `23x` `<` `28` `23x` `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. 
Question 3 of 4
3. Question
Solve for `x``61/2 x+5<6`Hint
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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`61/2 x+5` `<` `6` `61/2 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`21/3 y43>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, convert the equation to its standard form`21/3 y43` `>` `7` `21/3 y43` `+3` `>` `7` `+3` Add `3` to both sides `21/3 y4` `>` `10` `21/3 y4` `div2` `>` `10` `div2` Divide both sides by `2` `1/3 y4` `>` `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 y4` `>` `5` `1/3 y4` `+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 y4` `<` `5` `1/3 y4` `+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.
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