Solve and interpret a linear inequation

Inequality

\(\displaystyle 4x+1\le29\)

Which statement gives the solution?

Product A

\(\displaystyle A=2k\)

Product B

\(\displaystyle B=5k\)

Use the same multiplier for both products

How does A compare with B for positive and negative multipliers?

Keep track of the order

Use the inverse operations you used for equations. Keep the inequality sign until an operation requires it to reverse

1. 1Add or subtract on both sides The direction stays the same \(\displaystyle \begin{aligned}-5x+8&\ge23\\-5x+8-8&\ge23-8\\-5x&\ge15\end{aligned}\)
2. 2Divide by a negative Reverse the sign when dividing both sides by a negative number \(\displaystyle \begin{aligned}\frac{-5x}{-5}&\le\frac{15}{-5}\\x&\le-3\end{aligned}\)
3. 3Decide whether the boundary counts Less than or equal to includes equality. Strict less-than and greater-than signs exclude equality \(\displaystyle -5(-3)+8=23\)
4. 4Check a value on the chosen side A value smaller than the boundary satisfies the original inequality \(\displaystyle -5(-4)+8=28\ge23\)
5. 5Read reversed sides carefully Swapping the two sides changes how the comparison is written, but keeps the same meaning \(\displaystyle 9>x\quad\Longleftrightarrow\quad x<9\)

Reuse the equation methods

The denominator, bracket and collection steps from equations also apply to inequalities. Keep the comparison valid on both sides. Multiplying or dividing by a negative number reverses it

1. 1Clear a known positive denominator Multiply both sides by 2, add 4 to both sides, then divide by 3. Both factors are positive, so the sign stays the same \(\displaystyle \begin{aligned}\frac{3x-4}{2}&\ge7\\3x-4&\ge14\\3x&\ge18\\x&\ge6\end{aligned}\)
2. 2Undo a signed bracket factor Here divide both sides by negative 2 and reverse the sign, then subtract 3. Expanding the bracket first is also valid \(\displaystyle \begin{aligned}-2(x+3)&<8\\x+3&>-4\\x&>-7\end{aligned}\)
3. 3Collect unknown terms on both sides Subtract twice the unknown from both sides, add four to both sides, then divide by positive three \(\displaystyle \begin{aligned}5x-4&\le2x+8\\3x-4&\le8\\3x&\le12\\x&\le4\end{aligned}\)

The shading continues to the left beyond the displayed window. Which solution set does this line show?

Inequality

\(\displaystyle 3x+2<26\)

What is the greatest integer solution?

Isolate the unknown, then check whether the boundary itself is allowed

Inequality

\(\displaystyle 13-4x\le-17\)

What is the smallest integer solution?

Watch the inequality sign when you divide by the negative coefficient

Which line first changes the solution set?