Linear Inequalities - Algebra
Card 0 of 864
Which of the following inequalities is graphed above?
Which of the following inequalities is graphed above?
First, we determine the equation of the boundary line. This line includes points 
and 
, so the slope can be calculated as follows:
Since we also know the 
-intercept is 
, we can substitute 
in the slope-intercept form to obtain the equation of the boundary line:
The boundary is included, as is indicated by the line being solid, so the equality symbol is replaced by either 
or 
. To find out which one, we can test a point in the solution set - for ease, we will choose 
:
_____ 
_____ 
_____ 
0 is less than 3 so the correct symbol is 
.
The inequality is 
.
First, we determine the equation of the boundary line. This line includes points
Since we also know the
The boundary is included, as is indicated by the line being solid, so the equality symbol is replaced by either
0 is less than 3 so the correct symbol is
The inequality is
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Write an inequality for the following statement:
Two times the sum of 
and 
is greater than half of the product of 
and 
.
Write an inequality for the following statement:
Two times the sum of
When writing statments as inequalities, you must take it one step at a time. So in the statement
Two times the sum of 
and 
is greater than half of the product of 
and 
.
we will rewrite it one part at a time.
Two times the sum of 
and 
is greater than half of the product of 
and 
.
times the sum of 
and 
is greater than half of the product of 
and 
.
the sum of 
and 
is greater than half of the product of 
and 
.
is greater than half of the product of 
and 
.
half of the product of 
and 
.
of the product of 
and 
.
Simplify.
When writing statments as inequalities, you must take it one step at a time. So in the statement
Two times the sum of
we will rewrite it one part at a time.
Two times the sum of
Simplify.
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Write the inequality: Seven more than a number is less than five.
Write the inequality: Seven more than a number is less than five.
Split the statement into parts.
Seven more than a number: 
Is less than five: 
Combine both parts to form an inequality.
The answer is: 
Split the statement into parts.
Seven more than a number:
Is less than five:
Combine both parts to form an inequality.
The answer is:
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Solve the inequality:
Solve the inequality:
The inequality compares an absolute value function with a negative integer. Since the absolute value of any real number is greater than or equal to 0, it can never be less than a negative number. Therefore, 
can never happen. There is no solution.
The inequality compares an absolute value function with a negative integer. Since the absolute value of any real number is greater than or equal to 0, it can never be less than a negative number. Therefore,
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Solve this inequality.
Solve this inequality.
Split the inequality into two possible cases as follows, based on the absolute values.
First case: 
Second case: 
Let's find the inequality of the first case.
Multiply both sides by x + 6.
Subtract x from both sides, then subtract 3 from both sides.
Divide both sides by 3.
Let's find the inequality of the second case.
Multiply both sides by x + 6.
Simplify.
Add x to both sides, then subtract 3 from both sides.
Divide both sides by 5.
So the range of x-values is 
and 
.
Split the inequality into two possible cases as follows, based on the absolute values.
First case:
Second case:
Let's find the inequality of the first case.
Multiply both sides by x + 6.
Subtract x from both sides, then subtract 3 from both sides.
Divide both sides by 3.
Let's find the inequality of the second case.
Multiply both sides by x + 6.
Simplify.
Add x to both sides, then subtract 3 from both sides.
Divide both sides by 5.
So the range of x-values is
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Solve for 
:
Solve for
Solve for positive values by ignoring the absolute value. Solve for negative values by switching the inequality and adding a negative sign to 7.
Solve for positive values by ignoring the absolute value. Solve for negative values by switching the inequality and adding a negative sign to 7.
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Give the solution set for the following equation:
Give the solution set for the following equation:
First, subtract 5 from both sides to get the absolute value expression alone.
Split this into two linear equations:
or
The solution set is 
First, subtract 5 from both sides to get the absolute value expression alone.
Split this into two linear equations:
or
The solution set is
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Solve for 
in the inequality below.
Solve for
The absolute value gives two problems to solve. Remember to switch the "less than" to "greater than" when comparing the negative term.
or 
Solve each inequality separately by adding 
to all sides.
or 
This can be simplified to the format 
.
The absolute value gives two problems to solve. Remember to switch the "less than" to "greater than" when comparing the negative term.
Solve each inequality separately by adding
This can be simplified to the format
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Solve the inequality.
Solve the inequality.
Remove the absolute value by setting the term equal to either 
or 
. Remember to flip the inequality for the negative term!
Solve each scenario independently by subtracting 
from both sides.
Remove the absolute value by setting the term equal to either
Solve each scenario independently by subtracting
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Solve for 
:
Solve for
The absolute value of any number is nonnegative, so 
must always be greater than 
. Therefore, any value of 
makes this a true statement.
The absolute value of any number is nonnegative, so
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Solve for 
:
Solve for
The absolute value of a number must always be nonnegative, so 
can never be less than 
. This means the inequality has no solution.
The absolute value of a number must always be nonnegative, so
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Solve the inequality 
.
Solve the inequality
First, we can simplify this inequality by subtracting 7 from both sides. This gives us
Next, however, we need to make two separate inequalities due to the presence of an absolute value expression. What this inequality actually means is that
and
(Be careful with the inequality signs here! The second sign must be switched to allow for the effect of absolute value on negative numbers. In other words, the inequality must be greater than 
because, after the absolute value is applied, it will be less than 7.) When we solve the two inequalities, we get two solutions:
and
For the original statement to be true, both of these inequalities must be fulfilled. We're left with a final answer of
First, we can simplify this inequality by subtracting 7 from both sides. This gives us
Next, however, we need to make two separate inequalities due to the presence of an absolute value expression. What this inequality actually means is that
and
(Be careful with the inequality signs here! The second sign must be switched to allow for the effect of absolute value on negative numbers. In other words, the inequality must be greater than
and
For the original statement to be true, both of these inequalities must be fulfilled. We're left with a final answer of
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Solve the following inequality.
Solve the following inequality.
Inequalities involving 
generate two separate inequalities and can't be combined into a single inequality.
First isolate the absolute value expression on one side of the inequality
Subtract eight from each side.
From here separate the expression into two expressions for which we will need to solve.
Add six to get side.
Inequalities involving
First isolate the absolute value expression on one side of the inequality
Subtract eight from each side.
From here separate the expression into two expressions for which we will need to solve.
Add six to get side.
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Solve the inequality for 
.
Solve the inequality for
To solve an inequality with absolute value you have to consider the two equations it creates.
becomes 
and 
Solve for both inequalities by following the balancing rules. Be careful of division or multiplication of a negative number; if that happens, flip the inequality sign.
--
To solve an inequality with absolute value you have to consider the two equations it creates.
Solve for both inequalities by following the balancing rules. Be careful of division or multiplication of a negative number; if that happens, flip the inequality sign.
--
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Break the inequality into two parts:
For (2), because we multiplied one side by -1 we must flip the inequality sign. In general, when you multiply one side of an inequality by a negative number, the sign must be flipped.
AND
Final Solution 
Check with 
Break the inequality into two parts:
For (2), because we multiplied one side by -1 we must flip the inequality sign. In general, when you multiply one side of an inequality by a negative number, the sign must be flipped.
AND
Final Solution
Check with
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Which of the following graphs correctly depicts the graph of the inequality 
Which of the following graphs correctly depicts the graph of the inequality
Let's start by looking at the given equation:
The inequality is written in slope-intercept form; therefore, the slope is equal to 
and the y-intercept is equal to 
.
All of the graphs depict a line with slope of 
and y-intercept 
. Next, we need to decide if we should shade above or below the line. To do this, we can determine if the statement is true using the origin 
. If the origin satisfies the inequality, we will know to shade below the line. Substitute the values into the given equation and solve.
Because this statement is true, the origin must be included in the shaded region, so we shade below the line.
Finally, a statement that is "less than" or "greater than" requires a dashed line in the graph. On the other hand, those that are "greater than or equal to" or "less than or equal to" require a solid line. We will select the graph with shading below a dashed line.
Let's start by looking at the given equation:
The inequality is written in slope-intercept form; therefore, the slope is equal to
All of the graphs depict a line with slope of
Because this statement is true, the origin must be included in the shaded region, so we shade below the line.
Finally, a statement that is "less than" or "greater than" requires a dashed line in the graph. On the other hand, those that are "greater than or equal to" or "less than or equal to" require a solid line. We will select the graph with shading below a dashed line.
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We first need to eliminate the absolute value sign by making two inequalities:
and 
. Remember that when the number becomes negative we must flip the inequality symbol.
From there, it is a simple one-step inequality. We divide both sides by 
to get:
.
Because x lies between 
and 
, we can combine these into one inequality:
We first need to eliminate the absolute value sign by making two inequalities:
From there, it is a simple one-step inequality. We divide both sides by
Because x lies between
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Solve the inequality: 
Solve the inequality:
Split up the absolute value and rewrite the inequalities in two formats:
Solve the first inequality. Subtract three on both sides.
Divide by two on both sides.
The first solution is: 
Evaluate the second inequality.
Divide negative one on both sides. Dividing a negative term will switch the sign.
Subtract three on both sides.
Divide by two on both sides.
The second solution is: 
In interval notation, the answer is: 
Split up the absolute value and rewrite the inequalities in two formats:
Solve the first inequality. Subtract three on both sides.
Divide by two on both sides.
The first solution is:
Evaluate the second inequality.
Divide negative one on both sides. Dividing a negative term will switch the sign.
Subtract three on both sides.
Divide by two on both sides.
The second solution is:
In interval notation, the answer is:
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Solve: 
Solve:
Solve by first dividing three on both sides. We will need to isolate the absolute value sign before splitting the absolute value into its positive and negative components.
Split the absolute value into two equations.
Solve the first equation. Subtract three on both sides and simplify.
Divide by three on both sides.
This is the first solution. Solve the second equation. Divide by a negative one on both sides. This will switch the inequality sign.
Subtract three from both sides and simplify.
Divide by three on both sides.
The answer is between negative five and three but does not include both numbers: 
Solve by first dividing three on both sides. We will need to isolate the absolute value sign before splitting the absolute value into its positive and negative components.
Split the absolute value into two equations.
Solve the first equation. Subtract three on both sides and simplify.
Divide by three on both sides.
This is the first solution. Solve the second equation. Divide by a negative one on both sides. This will switch the inequality sign.
Subtract three from both sides and simplify.
Divide by three on both sides.
The answer is between negative five and three but does not include both numbers:
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