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Solving an inequality means finding the value or values that make the inequality true. In other words, finding the solution set. This can be done, similar to solving equations, by using inverse operations.

Solve the following inequality and graph its solution set. $x+3\leq 7$

Show Solution

To solve the inequality, we must isolate $x.$ Notice that the inverse of adding $3$ is subtracting $3.$
The inequality has the solutions $x\leq 4.$ To graph this on a number line, we can place a closed circle at $4,$ because $x$ can equal $4.$ Additionally, since $x$ can take all values less than $4,$ we shade the region to the left.

There is one important difference between solving equations and solving inequalities. When dividing or multiplying the inequality by a **negative** number, the inequality symbol must be **reversed** or **flipped**.

Consider the following example to understand why the inequality symbol is reversed. The inequality $2 \ < \ \ 5$ is a true statement. Multiplying both sides by $\text{-}1$ results in $\text{-} 2 \ \ \fbox{\phantom{>}} \ \ \text{-} 5.$

Notice that $\text{-} 5$ is smaller than $\text{-} 2.$ Thus, the inequality symbol that makes this a true statement is $>.$ Therefore, when multiplying or dividing an inequality by a negative number, the inequality symbol must be reversed.Solve the following inequality and graph its solution set. $\text{-} 2x>6$

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To isolate $x$ we can divide both sides of the inequality by $\text{-} 2.$
The solutions $x< \text{-} 3$ satisfy the inequality. We can graph the solution set on a number line. Since $x$ cannot equal $\text{-} 3,$ we place an open circle at $\text{-} 3.$ Additionally, to show all values **less than** $\text{-} 3,$ we'll shade the region to the left.

$\text{-} 2x>6$

DivNegIneqDivide by $\text{-} 2$ and flip inequality sign

$\dfrac{\text{-} 2x}{\text{-} 2}<\dfrac{6}{\text{-} 2}$

SimpQuotSimplify quotient

$x<\text{-} 3$

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