Solving One-Step Inequalities

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Algebra 1

One-Variable Inequalities

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Solving One-Step Inequalities 1.4 - Solution

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a

Inequalities can be solved in the same way as equations, by performing inverse operations on both sides until the variable is isolated. The only difference is that when you divide or multiply by a negative number, you must reverse the inequality sign. Here, we'll subtract both sides by $4$ so the inequality sign doesn't change.

$r+4 < 5$

SubIneq$\text{LHS}-4<\text{RHS}-4$

$r < 1$

This expression tells us that all values less than $1$ will satisfy the inequality. Note that $r$ cannot be equal to $1,$ which we show with an open circle on the number line.

b

We solve the inequality by using inverse operations on both sides until the $w$ is isolated.

$9+w > 7$

SubIneq$\text{LHS}-9>\text{RHS}-9$

$w > \text{-} 2$

This inequality tells us that all values greater than $\text{-} 2$ are solutions. Note that $w$ cannot equal $\text{-} 2,$ which we show with an open circle on the number line.

c

The variable $m$ can be isolated by subtracting $11$ from both sides of the inequality.

$11+m\geq15$

SubIneq$\text{LHS}-11\geq\text{RHS}-11$

$m\geq4$

This expression tells us that all values greater than or equal to $4$ will satisfy the inequality. Since $m$ can equal $4,$ we will mark it as a closed circle on the number line.

d

To solve the inequality we'll subtract $w$ from both sides to gather all $w$-terms.

$w-5\leq2w$

SubIneq$\text{LHS}-w\leq\text{RHS}-w$

$\text{-}5\leq w$

RearrangeIneqRearrange inequality

$w\geq\text{-}5$

This inequality tells us that all values greater than or equal to $\text{-} 5$ will satisfy the inequality. Thus, $\text{-}5$ should be marked as an closed circle and the part to the right represents all values greater than $\text{-}5.$