Linear Inequalities in One Variable
Learning Objectives
- Use the addition and multiplication properties to solve algebraic inequalities
- Express solutions to inequalities graphically, with interval notation, and as an inequality
- Simplify and solve algebraic inequalities using the distributive property to clear parentheses and fractions
Using the Properties of Inequalities
When we work with inequalities, we can usually treat them similarly to but not exactly as we treat equalities. We can use the addition property and the multiplication property to help us solve them. The one exception is when we multiply or divide by a negative number; doing so reverses the inequality symbol. There are three ways to represent solutions to inequalities: an interval, a graph, and an inequality. Because there is usually more than one solution to an inequality, when you check your answer you should check the end point and one other value to check the direction of the inequality. When we work with inequalities, we can usually treat them similarly to but not exactly as we treat equalities.A General Note: Properties of Inequalities
[latex]\begin{array}{ll}\text{Addition Property}\hfill& \text{If }a< b,\text{ then }a+c< b+c.\hfill \\ \hfill & \hfill \\ \text{Multiplication Property}\hfill & \text{If }a< b\text{ and }c> 0,\text{ then }ac< bc.\hfill \\ \hfill & \text{If }a< b\text{ and }c< 0,\text{ then }ac> bc.\hfill \end{array}[/latex]
These properties also apply to [latex]a\le b[/latex], [latex]a>b[/latex], and [latex]a\ge b[/latex].Start With | Multiply By | Final Inequality |
[latex]a>b[/latex] | [latex]c[/latex] | [latex]ac>bc[/latex] |
[latex]5>3[/latex] | [latex]3[/latex] | [latex]15>9[/latex] |
[latex]a>b[/latex] | [latex]-c[/latex] | [latex]-ac<-bc[/latex] |
[latex]5>3[/latex] | [latex]-3[/latex] | [latex]-15<-9[/latex] |
Start With | Divide By | Final Inequality |
[latex]a>b[/latex] | [latex]c[/latex] | [latex] \displaystyle \frac{a}{c}>\frac{b}{c}[/latex] |
[latex]4>2[/latex] | [latex]2[/latex] | [latex] \displaystyle \frac{4}{2}>\frac{2}{2}[/latex] |
[latex]a>b[/latex] | [latex]-c[/latex] | [latex] \displaystyle -\frac{a}{c}<-\frac{b}{c}[/latex] |
[latex]4>2[/latex] | [latex]-2[/latex] | [latex] \displaystyle -\frac{4}{2}<-\frac{2}{2}[/latex] |
Example: Demonstrating the Addition Property
Illustrate the addition property for inequalities by solving each of the following:- [latex]x - 15<4[/latex]
- [latex]6\ge x - 1[/latex]
- [latex]x+7>9[/latex]
Answer: The addition property for inequalities states that if an inequality exists, adding or subtracting the same number on both sides does not change the inequality.
- [latex]\begin{array}{ll}x - 15<4\hfill & \hfill \\ x - 15+15<4+15 \hfill & \text{Add 15 to both sides.}\hfill \\ x<19\hfill & \hfill \end{array}[/latex]
- [latex]\begin{array}{ll}6\ge x - 1\hfill & \hfill \\ 6+1\ge x - 1+1\hfill & \text{Add 1 to both sides}.\hfill \\ 7\ge x\hfill & \hfill \end{array}[/latex]
- [latex]\begin{array}{ll}x+7>9\hfill & \hfill \\ x+7 - 7>9 - 7\hfill & \text{Subtract 7 from both sides}.\hfill \\ x>2\hfill & \hfill \end{array}[/latex]
Try It
Solve [latex]3x - 2<1[/latex].Answer: [latex-display]x<1[/latex-display]
Example: Demonstrating the Multiplication Property
Illustrate the multiplication property for inequalities by solving each of the following:- [latex]3x<6[/latex]
- [latex]-2x - 1\ge 5[/latex]
- [latex]5-x>10[/latex]
Answer:
- [latex]\begin{array}{l}3x<6\hfill \\ \frac{1}{3}\left(3x\right)<\left(6\right)\frac{1}{3}\hfill \\ x<2\hfill \end{array}[/latex]
- [latex]\begin{array}{ll}-2x - 1\ge 5\hfill & \hfill \\ -2x\ge 6\hfill & \hfill \\ \left(-\frac{1}{2}\right)\left(-2x\right)\ge \left(6\right)\left(-\frac{1}{2}\right)\hfill & \text{Multiply by }-\frac{1}{2}.\hfill \\ x\le -3\hfill & \text{Reverse the inequality}.\hfill \end{array}[/latex]
- [latex]\begin{array}{ll}5-x>10\hfill & \hfill \\ -x>5\hfill & \hfill \\ \left(-1\right)\left(-x\right)>\left(5\right)\left(-1\right)\hfill & \text{Multiply by }-1.\hfill \\ x<-5\hfill & \text{Reverse the inequality}.\hfill \end{array}[/latex]
Try It
Solve [latex]4x+7\ge 2x - 3[/latex].Answer: [latex]x\ge -5[/latex]
Solving Inequalities in One Variable Algebraically
As the examples have shown, we can perform the same operations on both sides of an inequality, just as we do with equations; we combine like terms and perform operations. To solve, we isolate the variable.Example: Solving an Inequality Algebraically
Solve the inequality: [latex]13 - 7x\ge 10x - 4[/latex].Answer: Solving this inequality is similar to solving an equation up until the last step.
Try It
Solve the inequality and write the answer using interval notation: [latex]-x+4<\frac{1}{2}x+1[/latex].Answer: [latex]\left(2,\infty \right)[/latex]
Example: Solving an Inequality with Fractions
Solve the following inequality and write the answer in interval notation: [latex]-\frac{3}{4}x\ge -\frac{5}{8}+\frac{2}{3}x[/latex].Answer: We begin solving in the same way we do when solving an equation.
Try It
Solve the inequality and write the answer in interval notation: [latex]-\frac{5}{6}x\le \frac{3}{4}+\frac{8}{3}x[/latex].Answer: [latex-display]\left[-\frac{3}{14},\infty \right)[/latex-display]
Simplify and solve algebraic inequalities using the distributive property
As with equations, the distributive property can be applied to simplify expressions that are part of an inequality. Once the parentheses have been cleared, solving the inequality will be straightforward.Example
Solve for x. [latex]2\left(3x–5\right)\leq 4x+6[/latex]Answer: Distribute to clear the parentheses.
[latex] \displaystyle \begin{array}{r}\,2(3x-5)\leq 4x+6\\\,\,\,\,6x-10\leq 4x+6\end{array}[/latex]
Subtract 4x from both sides to get the variable term on one side only.[latex]\begin{array}{r}6x-10\le 4x+6\\\underline{-4x\,\,\,\,\,\,\,\,\,\,\,\,\,\,-4x}\,\,\,\,\,\,\,\,\,\\\,\,\,2x-10\,\,\leq \,\,\,\,\,\,\,\,\,\,\,\,6\end{array}[/latex]
Add 10 to both sides to isolate the variable.[latex]\begin{array}{r}\\\,\,\,2x-10\,\,\le \,\,\,\,\,\,\,\,6\,\,\,\\\underline{\,\,\,\,\,\,+10\,\,\,\,\,\,\,\,\,+10}\\\,\,\,2x\,\,\,\,\,\,\,\,\,\,\,\le \,\,\,\,\,16\,\,\,\end{array}[/latex]
Divide both sides by 2 to express the variable with a coefficient of 1.[latex]\begin{array}{r}\underline{2x}\le \,\,\,\underline{16}\\\,\,\,2\,\,\,\,\,\,\,\,\,\,\,\,\,2\,\,\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x\,\,\,\le \,\,\,\,\,8\end{array}[/latex]
Answer
Inequality: [latex]x\le8[/latex] Interval: [latex]\left(-\infty,8\right][/latex] Graph: The graph of this solution set includes 8 and everything left of 8 on the number line. Check the solution.Answer: First, check the end point 8 in the related equation.
[latex] \displaystyle \begin{array}{r}2(3x-5)=4x+6\,\,\,\,\,\,\\2(3\,\cdot \,8-5)=4\,\cdot \,8+6\\\,\,\,\,\,\,\,\,\,\,\,2(24-5)=32+6\,\,\,\,\,\,\\2(19)=38\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\\38=38\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\end{array}[/latex]
Then, choose another solution and evaluate the inequality for that value to make sure it is a true statement. Try 0.[latex] \displaystyle \begin{array}{l}2(3\,\cdot \,0-5)\le 4\,\cdot \,0+6?\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2(-5)\le 6\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-10\le 6\,\,\end{array}[/latex]
[latex-display]x\le8[/latex] is the solution to [latex]\left(-\infty,8\right][/latex-display]Try It
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- Solve a Linear Inequality Requiring Multiple Steps (One Var). Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.