Solving One-Step Inequalities

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Algebra 1

One-Variable Inequalities

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Solving One-Step Inequalities 1.5 - 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.

$\text{-}3y\leq\text{-}9$

DivNegIneqDivide by $\text{-}3$ and flip inequality sign

$y\geq3$

This inequality tells us that all values greater than or equal to $\text{-}3$ are solutions. Note that $y$ can equal $\text{-}3,$ which we show with a closed circle on the number line.

b

To isolate $a,$ we multiply both sides by $4.$

$\dfrac{a}{4}<10.2$

MultIneq$\text{LHS}\cdot 4 < \text{RHS} \cdot 4$

$a<40.8$

This expression tells us that all values less than $40.8$ will satisfy the inequality. This can be graphed with an open circle at $40.8$ and a shaded region to the left.

c

If we pretend that the inequality is an equation, $n$ can be isolated if we multiply both sides by $\text{-}3.$ We can do that for the inequality too, but must remember to flip the inequality sign.

$\dfrac{n}{\text{-}3}\geq1$

MultNegIneqMultiply by $\text{-} 3$ and flip inequality sign

$n\leq\text{-}3$

This expression tells us that all values less than or equal to $\text{-}3$ will satisfy the inequality. Note that since $n$ can equal $\text{-}3,$ it's shown with a closed circle on the number line.

d

Since the inverse operation of division is multiplication, we should multiply both sides by $4.$

$\dfrac{1}{4}m \leq \text{-}17$

MoveRightFacToNumOne$\dfrac{1}{b}\cdot a = \dfrac{a}{b}$

$\dfrac{m}{4} \leq \text{-}17$

MultIneq$\text{LHS}\cdot 4 \leq \text{RHS} \cdot 4$

$m\leq \text{-} 68$

The inequality should be graphed with a closed circle at $\text{-}68$ and a shaded region to the left on the number line.