Solving Inequalities Algebraically

Visualizing inequalities and their solutions

We will start by looking at the inequality $x+1 < 3$. This inequality, translated into words, becomes “$x+1$ is less than 3.” So we’ll start by looking at the graph for $x+1$ ($y=x+1$) and the graph for being less than 3 ($y < 3$). These two graphs are shown to the left.

At which values of $x$ do these two graphs overlap? In other words, for which $x$ values is the red line inside the blue region?

Choose one specific value of $x$ where the red line is inside the blue region.

Check to see if this value of $x$ is a solution to the inequality $x+1 < 3$.

Solving $x+b < c$ and related inequalities

In an earlier lesson, we saw that you could picture an equation by graphing its left-hand side and its right-hand side to see what $x$-value(s) make the sides equal. Here we are picturing an inequality in a similar way. We graph its left-hand side, and also graph the rest of the inequality, which is the condition or rule that must be true of the left-hand side for the entire inequality to be true. This means that the $x$-values where the two graphs overlap are the $x$-values that make the original inequality true. In other words, they are the solutions of the inequality.

We will continue looking at the inequality $x+1 < 3$. However, there is now also a slider to the left which adds the same number $k$ to both sides of the inequality $x+1 < 3$, changing it to $x+1+k < 3+k$. This means that the grid to the left will show the graphs of $y=x+1+k$ and $y < 3+k$.

Slide the slider to $k=-4$. At which values of $x$ do the two graphs overlap?

When $k=-4$, notice that the inequality can be simplified from $x+1-4 < 3-4$ to $x-3 < -1$.

Slide the slider to $k=-2$. At which values of $x$ do the two graphs overlap?

What is the simplified form of the inequality $x+1+k < 3+k$ when $k=-2$?

As you slide the slider, is there any change in the values of $x$ where the two graphs overlap?

Slide the slider to $k=-1$. What is the simplified form of the inequality $x+1+k < 3+k$ now?

As you have seen both in the previous lesson and in addBothSides, adding the same number to both sides of an inequality doesn’t change when that inequality is true. You can use this fact to solve simple inequalities.

Each row of the table has an inequality in it. Slide the $k$ slider to solve that inequality. Then pick some $x$ in the solution set you found, and check that it is a solution to the original inequality.

Solving $ax < c$ and related inequalities

We will now picture the inequality $2x > 3$. As before, we do this by graphing the equation $y=2x$ and the inequality $y > 3$ on the same coordinate grid, and seeing where they overlap.

The $m$ slider multiplies both sides of the inequality by the same positive number. Does sliding the $m$ slider change the $x$ values where the inequality is true (that is, where the graphs overlap)?

Set $m=0.5$. The inequality pictured on the left is now $0.5(2x) > 0.5(3)$. What is the simplified form of this inequality?

To solve the inequality, you set $m=0.5$. Write 0.5 as a fraction.

The grid on the left now shows the inequality $-2x≥1$, and the $m$ slider has been extended to include negative values. As you saw in the last lesson, multiplying an inequality by a negative number requires you to change the direction of the inequality. Because of this, sliding the $m$ slider to a negative number also changes the direction of the inequality pictured.

Slide the slider to $m=-1$. What is the inequality pictured to the left (in simplified form)?

Does sliding the $m$ slider to any nonzero value (positive or negative) change the $x$ values where the inequality is true?

If you want to solve this inequality (get $x$ alone on one side), what should you slide $m$ to?

Write the $m$ value you just found as a fraction.

Solve the inequality.

Notice how, when you were solving an inequality where $x$ was multiplied by $-2$, you multiplied both sides of the inequality by $$1/{-2}$$ (just as you would if you were solving an equation). In general, if you have an equation or inequality where $x$ is multiplied by $a$, you can solve it by multiplying by $$1/a$$.

Each row of this table has an inequality in it. Solve that inequality. Then pick some $x$ in the solution set, and check that it is a solution to the original inequality.

Solving $ax+b < c$ and related inequalities

The inequality $-2x-1≥3$ is pictured to the left, along with a slider that adds $k$ to both sides of the inequality.

What number $k$ would you have to add to both sides if you wanted to make this inequality simpler? (We want to make its left-hand side simpler, so we can solve for $x$.)

What would that simpler inequality be?

Click to see a graph of that simpler inequality, along with a slider that multiplies both sides of the inequality by $m$.

What number would you have to multiply both sides by if you wanted to solve this inequality?

Solve this inequality.

Check some number in this solution set.

Each row of the table below gives an inequality. Solve each inequality and check some solution. (Remember to reduce any fractions to lowest terms.)